OBSERVATION OF MUON NEUTRINO CHARGED ...lss.fnal.gov/archive/thesis/2000/fermilab-thesis-2014-24.pdf2014 ABSTRACT OBSERVATION OF MUON NEUTRINO CHARGED CURRENT EVENTS IN AN OFF-AXIS

OBSERVATION OF MUON NEUTRINO CHARGED CURRENT EVENTS IN ANOFF-AXIS HORN-FOCUSED NEUTRINO BEAM USING THE NOνA PROTOTYPE

DETECTOR

By

Enrique Arrieta Dıaz

A DISSERTATION

Submitted toMichigan State University

in partial fulfillment of the requirementsfor the degree of

Physics - Doctor of Philosophy

2014

ABSTRACT

OBSERVATION OF MUON NEUTRINO CHARGED CURRENT EVENTSIN AN OFF-AXIS HORN-FOCUSED NEUTRINO BEAM USING THE

NOνA PROTOTYPE DETECTOR

By

Enrique Arrieta Dıaz

The NOνA is a long base-line neutrino oscillation experiment. It will study the oscil-

lations between muon and electron neutrinos through the Earth. NOνA consists of two

detectors separated by 810 km. Each detector will measure the electron neutrino content of

the neutrino (NuMI) beam. Differences between the measurements will reveal details about

the oscillation channel. The NOνA collaboration built a prototype detector on the surface

at Fermilab in order to develop calibration, simulation, and reconstruction tools, using real

data. This 220 ton detector is 110 mrad off the NuMI beam axis. This off-axis location

allows the observation of neutrino interactions with energies around 2 GeV, where neutrinos

come predominantly from charged kaon decays. During the period between October 2011

and April 2012, the prototype detector collected neutrino data from 1.67 × 1020 protons on

target delivered by the NuMI beam. This analysis selected a number of candidate charged

current muon neutrino events from the prototype data, which is 30% lower than predicted by

the NOνA Monte Carlo simulation. The analysis suggests that the discrepancy comes from

an over estimation of the neutrino flux in the Monte Carlo simulation, and in particular,

from neutrinos generated in charged kaon decays. The ratio of measured divided by the sim-

ulated flux of muon neutrinos coming from charged kaon decays is: 0.70+0.108−0.094. The NOνA

collaboration may use the findings of this analysis to introduce a more accurate prediction

of the neutrino flux produced by the NuMI beam in future Monte Carlo simulations.

To my abuelo, Andres, Quique and Rochi.The precious components of my high intensity beam of unconditional love.

iii

ACKNOWLEDGMENTS

I can not pinpoint a moment in my life as the time when I decided to be a scientist. However,

I can assure that my decision consolidated during my third year of elementary school in the

classroom of C. Cabezas. He impacted my life forever, and my incommensurable gratitude

accompanies him now in the stars, right where he belongs. Soon after I learned from M. Rios

that a successful man can only live under the strict regime of impeccable moral standards,

and I have tried to follow his teachings to the best of my capacities. During my junior and

senior years of high school I sat in the classroom of C. Martınez where he elegantly provided

most of the knowledge in physics that I bear today. It was at this time when I told F. Casas

that if by 2007 I was not a rey de reyes in Valledupar, he could be certain that I would then

become a Ph.D. in physics. Thanks to him I was introduced to the books of Carl Sagan that

structured my scientific thinking. Thanks to all my teachers at Abraham Lincoln school,

and to my professors at FIT, Los Andes, and MSU for their wonderful lessons.

The long road that brought me to my present had many gratifying moments, and count-

less steep slopes and deep troughs. This road, that resembled that to Santa Ana during

heavy winter times, felt very smooth thanks to the love and support of those members of my

families Aaron, Arrieta, Dıaz, and Hamburger, that always believed in me. I never tried to

make them feel proud, however I have been working extremely hard not to make them feel

ashamed. My moments of happiness were doubled, and my periods of sadness were halved,

when I had the chance to share them with my siblings from the 16th street and my fellow

Wachakeros. When the wipers did not work due to the mud covering the windshield, only

the GPS guidance of E. M. Aaron helped me to reach safe shelter. My personal oracle, A.

Op den Bosch, kept me safe and well advised. An especial gratefulness remains in my heart

iv

for the members of the Solano family that cared for me like one of their own. Many friends

have their two cents and their grains of sand deposited in the safe of my heart. I can not

name them all, but I do treasure all their support throughout the years.

I have felt like Achilles racing the tortoise many times. Nonetheless, there is always

someone out there to remind me that only a fool can not defeat a tortoise at a speed race,

and that I am not a fool. M. C. Jacome and M. Nowakowski made this very clear to me.

However, it is only with the help of extraordinary persons that extraordinary goals are

accomplished. I am lucky enough to be surrounded by extraordinary individuals like S. D.

Mahanti, J. Paley, and S. Pratt who supported me, beyond imagination, when I most needed

them. There is no word in human languages to express the infinite gratitude that I have for

them. My friendships with J. Clifford, E. Kessler, T. Olson, and L. C. Rodrıguez made of

my years at MSU a wonderful time.

The quest in NOνA started with the Fellowship of the Fiber, and my companions M.

Nila, R. Richards, and D. Shooltz. Days of darkness were illuminated by their amazing

personalities. Then I arrived to the land of Fermilab where I found the invaluable support

of M. Betancourt. After meeting her the locomotive became unstoppable. M. Sanchez and

P. Vahle provided endless fuel for my endeavor. My analysis was only possible with the

outstanding patronage of C. Backhouse, M. Baird, M. Frank, R. Hatcher, N. Mayer, M.

Messier, M. Muether, E. Niner, R. Patterson, G. Pawloski, N. Raddatz, P. Shanahan, and Z.

Wang. A. Sousa greatly contributed to my success with his knowledge about the divine and

human. Indubitably the entire NOνA collaboration made it happen. The last portion of my

journey, at Argonne Lab, was enlightened by M. Goodman, and S. Magill, R. Talaga, and the

members of the neutrino group. They made me feel like home. In particular, J. A. Sepulveda

showed me the true meaning of humbleness and kindness. I will be forever thankful to D.

v

Barratt, E. Johnson, B. Wenzlick, and R. Young. Their magic always simplified my life.

Mission accomplished! Mabel can now host the party, and Manolo can prepare the hug

and the entertainment. Leandro made El Cardon Guajiro, and Diomedes sang it for me.

The great days are straight ahead because I am, indeed, made for great things. Thank you!

“Hay soledades que duelen mucho, hay un silencio para pensar, hoy quiero luces para que

alumbren lo que me falta por caminar. Soy andariego de los caminos, con palabritas quiero

decir, ası es mi vida siempre cambiando, un dıa muy triste y un dıa feliz. Solo le pido a la

vida, que me de felicidad, y que mi conciencia duerma siempre tranquila. Quiero sentir el

aprecio de los amigos, vivir feliz en mi Valle, no pido mas. No pido mas, no quiero ser

golondrina que errante va. No pido mas, hoy quiero luces que alumbren mi oscuridad.”

Gustavo Gutierrez Cabello

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Brief History Of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Motivations For The NOνA Experiment . . . . . . . . . . . . . . . . . . . . 31.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2 Theoretical Framework for Neutrinos . . . . . . . . . . . . . . . . 92.1 Neutrinos In Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Charged Current Interactions . . . . . . . . . . . . . . . . . . . . . . 102.1.1.1 Quasi-elastic Scattering . . . . . . . . . . . . . . . . . . . . 112.1.1.2 Resonant Pion Production . . . . . . . . . . . . . . . . . . . 142.1.1.3 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . 162.1.1.4 Inclusive Cross Section Measurements . . . . . . . . . . . . 17

2.1.2 Neutral Current Interactions . . . . . . . . . . . . . . . . . . . . . . . 202.2 Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Seesaw Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Chapter 3 NuMI Beam Overview . . . . . . . . . . . . . . . . . . . . . . . . . 343.1 Primary Beam System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Secondary Beam System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 NuMI Beam Composition And Energy Spectrum At 110 mrad . . . . . . . . 41

Chapter 4 The NOνA Prototype Detector . . . . . . . . . . . . . . . . . . . . 474.1 Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 The Near Detector On The Surface . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Liquid Scintillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Wavelength Shifting Fiber . . . . . . . . . . . . . . . . . . . . . . . . 564.2.3 Avalanche Photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.4 Front-End Boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.5 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.6 Performance And Calibration Of The Prototype Detector . . . . . . . 63

4.3 NOνA Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.2 Reconstruction Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.2.1 The Slicer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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4.3.2.2 Track Reconstruction . . . . . . . . . . . . . . . . . . . . . . 734.4 Data Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.5 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Chapter 5 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.1 Charged And Neutral Current Neutrino Interactions . . . . . . . . . . . . . . 955.2 Event Containment Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Charged Current Event Selection . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.1 Longest Track Length Cut . . . . . . . . . . . . . . . . . . . . . . . . 1035.3.2 Cosmic Ray Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3.3 Minimum Ionizing Particle Cut . . . . . . . . . . . . . . . . . . . . . 109

Chapter 6 Event Energy Reconstruction . . . . . . . . . . . . . . . . . . . . . 1156.1 Muon Energy Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Hadronic Energy Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3 Quasi-elastic And Non-quasi-elastic Classification . . . . . . . . . . . . . . . 1256.4 Neutrino Energy Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Chapter 7 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 1377.1 Energy Estimation Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 1387.2 Prototype Detector’s Channels Configuration Uncertainty . . . . . . . . . . . 1417.3 GENIE Cross Sections And Final State Physics Uncertainty . . . . . . . . . 1427.4 Unfolding Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 1477.5 Total Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Chapter 8 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.1 Data And Monte Carlo Simulation Comparison . . . . . . . . . . . . . . . . 1528.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Chapter 9 Discussion And Final Remarks . . . . . . . . . . . . . . . . . . . . 1769.1 Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.1.1 Number Of Atoms In The Target Region . . . . . . . . . . . . . . . . 1809.1.2 Monte Carlo Cross Sections . . . . . . . . . . . . . . . . . . . . . . . 1829.1.3 Total Reconstruction Efficiency . . . . . . . . . . . . . . . . . . . . . 1849.1.4 νµ + νµ Flux Coming From Charged Kaon Decays . . . . . . . . . . 185

9.2 Inclusive νµ Charged Current Cross Section Calculation . . . . . . . . . . . . 1899.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Appendix A Mass Terms In The Weak Interaction . . . . . . . . . . . . . . . 197Appendix B NOνA Kalman Tracker . . . . . . . . . . . . . . . . . . . . . . . . 204Appendix C Unfolding Algorithm, TSVDUnfold . . . . . . . . . . . . . . . . 209Appendix D List Of Muon Neutrino Charged Current Candidate Events . 220

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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

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LIST OF TABLES

Table 2.1 Neutrino Oscillation Parameters. . . . . . . . . . . . . . . . . . . 32

Table 4.1 Kalman Tracker Performance. . . . . . . . . . . . . . . . . . . . . 74

Table 5.1 Event Selection Performance. . . . . . . . . . . . . . . . . . . . . 114

Table 7.1 GENIE Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Table 8.1 Event Selection Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . 159

Table 8.2 Quasi-elastic And Non-quasi-elastic Classification Performan-ce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Table 8.3 Candidate Events With Background. . . . . . . . . . . . . . . . . 173

Table 8.4 Number Of Candidate Events. . . . . . . . . . . . . . . . . . . . . 174

Table 8.5 Neutral Current Background. . . . . . . . . . . . . . . . . . . . . 174

Table 8.6 Electron Neutrino Background. . . . . . . . . . . . . . . . . . . . 175

Table 9.1 Background To Neutrinos From Kaon Decays. . . . . . . . . . . 178

Table 9.2 Candidate Events From Kaon Decays. . . . . . . . . . . . . . . . 179

Table 9.3 Chemical Composition Of The Prototype Detector. . . . . . . 181

Table D.1 Quasi-elastic Candidate Events. . . . . . . . . . . . . . . . . . . . 220

Table D.2 Non-quasi-elastic Candidate Events, I. . . . . . . . . . . . . . . . 221

Table D.3 Non-quasi-elastic Candidate Events, II. . . . . . . . . . . . . . . 222

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LIST OF FIGURES

Figure 2.1 Weak Interaction Charged Current Representation Of Neutri-no-nucleon Scattering. . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.2 Experimental Results On Muon Neutrino Quasi-elastic CrossSections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Figure 2.3 Theoretical And Experimental Results On Muon NeutrinoResonance Cross Sections. . . . . . . . . . . . . . . . . . . . . . 15

Figure 2.4 Experimental Results On The Total Inclusive Muon Neu-trino Charged Current Cross Sections. . . . . . . . . . . . . . 17

Figure 2.5 Brookhaven National Laboratory Muon Neutrino InclusiveCross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 2.6 SciBooNE Muon Neutrino Inclusive Cross Sections. . . . . . 19

Figure 2.7 Weak Interaction Neutral Current Representation Of Neutri-no-nucleon Scattering. . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 2.8 MiniBooNE Neutral Current Differential Cross Section. . . 22

Figure 2.9 Neutrino Mass Hierarchy. . . . . . . . . . . . . . . . . . . . . . . 33

Figure 3.1 Fermilab Accelerator Campus. . . . . . . . . . . . . . . . . . . . 34

Figure 3.2 NuMI Beam Trajectory. . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 3.3 NuMI Beam Energy Spectrum. . . . . . . . . . . . . . . . . . . 35

Figure 3.4 Location Of The NOνA And MINOS Far Detectors. . . . . . 36

Figure 3.5 Schematics Of The NuMI Complex. . . . . . . . . . . . . . . . 37

Figure 3.6 NuMI Beam Schematics. . . . . . . . . . . . . . . . . . . . . . . 37

Figure 3.7 NuMI Target Drawing. . . . . . . . . . . . . . . . . . . . . . . . 39

Figure 3.8 Beam And Electric Current Directions Through The NuMIHorns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xi

Figure 3.9 Meson Transverse And Longitudinal Momenta. . . . . . . . . 43

Figure 3.10 Neutrino Energy Distribution Discriminated By The MotherParticle Of The Neutrino. . . . . . . . . . . . . . . . . . . . . . 44

Figure 3.11 Neutrino Energy Distribution Discriminated By The Neu-trino Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 3.12 Ratio Of Neutrinos From Charged Kaon Decays Over Neu-trinos From Charged Pion Decays. . . . . . . . . . . . . . . . . 46

Figure 4.1 Electron Neutrino Bi-probability Plot. . . . . . . . . . . . . . 48

Figure 4.2 Drawing Of The NOνA Detectors. . . . . . . . . . . . . . . . . 50

Figure 4.3 Angle Of Neutrinos With Respect To The NuMI Beam. . . 51

Figure 4.4 Meson Momentum And Neutrino Energy As A Function OfThe NuMI Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure 4.5 Drawing Of The NOνA Prototype Detector. . . . . . . . . . . 53

Figure 4.6 NOνA Extrusions And Cells Sample. . . . . . . . . . . . . . . 54

Figure 4.7 Side View Of The NOνA Prototype Detector. . . . . . . . . . 54

Figure 4.8 Emission Spectra Of The Wavelength Shifting Fiber. . . . . 57

Figure 4.9 Wavelength Shifting Fiber Transverse Area. . . . . . . . . . . 58

Figure 4.10 Avalanche Photodiode. . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 4.11 Schematic Overview Of The Data Acquisition System. . . . 62

Figure 4.12 Event Display Of The NOνA Prototype Detector. . . . . . . 64

Figure 4.13 Cosmic Data Of The NOνA Prototype Detector: Light Out-put vs. Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 4.14 Energy Calibration Of The NOνA Prototype Detector. . . . 66

Figure 4.15 Energy Conversion Factors Of The NOνA Prototype Detec-tor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Figure 4.16 Light Level Over Time On The NOνA Prototype Detector. 69

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Figure 4.17 Sample Cosmic Ray Event For Slicer Performance. . . . . . . 72

Figure 4.18 Event Vertex Resolution. . . . . . . . . . . . . . . . . . . . . . . 75

Figure 4.19 Longest Track Endpoint Resolution. . . . . . . . . . . . . . . . 76

Figure 4.20 Sample Event Of Tracking Algorithm Corrections. Case 1. . 77

Figure 4.21 Sample Events Of Tracking Algorithm Corrections. Cases 2To 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Figure 4.22 Sample Event Of Tracking Algorithm Corrections. . . . . . . 80

Figure 4.23 Number Of Active Channels In The NOνA Prototype De-tector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Figure 4.24 Number Of Neutrino Candidates As A Function Of The Pro-tons On Target In The NOνA Prototype Detector. . . . . . . 83

Figure 4.25 Time Of Event Slice In The Data Of The NOνA PrototypeDetector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Figure 4.26 Sample Trigger Time Window With A Selected NeutrinoCandidate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 4.27 Time Of Slices In The Data Of The Prototype Detector AfterEvent Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 4.28 Sample Trigger Time Window With Cosmic Ray Events. . . 87

Figure 4.29 Sample Cosmic Ray Event. . . . . . . . . . . . . . . . . . . . . . 88

Figure 4.30 Beam To NOνA Prototype Detector Coordinate Transfor-mation Illustration. . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Figure 4.31 The Direction Of Neutrinos From The NuMI Beam In ThePrototype Detector. . . . . . . . . . . . . . . . . . . . . . . . . . 91

Figure 4.32 Longest Track Angular Distributions For Data And MonteCarlo Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 5.1 Single Particle Events. . . . . . . . . . . . . . . . . . . . . . . . . 94

Figure 5.2 Sample Events For Neutral Current Background. . . . . . . . 95

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Figure 5.3 Neutrino Energy Distributions From The NOνA PrototypeDetector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Figure 5.4 Monte Carlo Prediction Of The Ratio Of Charged CurrentTo Neutral Current Neutrino Interactions. . . . . . . . . . . . 97

Figure 5.5 Fraction of Reconstructed Charged Current Events. . . . . . 97

Figure 5.6 Ratio Of Reconstructed Over Simulated Events. . . . . . . . 98

Figure 5.7 Vertex Region Of The NOνA Prototype Detector. . . . . . . 100

Figure 5.8 Containment Region Of The NOνA Prototype Detector. . . 101

Figure 5.9 Monte Carlo Simulated Longest Track Length DistributionsFor Muons and Non-muons. . . . . . . . . . . . . . . . . . . . . 103

Figure 5.10 Monte Carlo Simulation Of The Longest Track Length Dis-tributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 5.11 Cosmic Rays Rejection Zone By The Cut In cos θY . . . . . . 106

Figure 5.12 Cosmic Rays Rejection Zone By The Cut In cos θNuMI. . . . 108

Figure 5.13 cos θY vs. cos θNuMI. . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 5.14 Cosmic Track Mean dE/dX. . . . . . . . . . . . . . . . . . . . . 109

Figure 5.15 Minimum Ionizing Particle Fraction. . . . . . . . . . . . . . . . 110

Figure 5.16 Longest Track Particle Identity. . . . . . . . . . . . . . . . . . . 111

Figure 5.17 Consequences Of The Minimum Ionizing Particle Cut. . . . 112

Figure 6.1 Muon: True Energy vs. Reconstructed Track Length. . . . . 116

Figure 6.2 Profile Plots Of Figure 6.1a. . . . . . . . . . . . . . . . . . . . . 118

Figure 6.3 Muon Energy Resolution. . . . . . . . . . . . . . . . . . . . . . . 119

Figure 6.4 Muon Energy Distributions. . . . . . . . . . . . . . . . . . . . . 120

Figure 6.5 Hadronic Energy Classification Zones. . . . . . . . . . . . . . . 122

Figure 6.6 True Hadronic Energy vs. Deposited Energy. . . . . . . . . . 123

xiv

Figure 6.7 Hadronic Energy Resolution. . . . . . . . . . . . . . . . . . . . 124

Figure 6.8 Hadronic Energy Distributions. . . . . . . . . . . . . . . . . . . 125

Figure 6.9 Quasi-elastic And Non-quasi-elastic Hadronic Energy Dis-tributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Figure 6.10 Muon And Hadronic Energy Distributions For The Quasi-elastic And Non-quasi-elastic Events. . . . . . . . . . . . . . . 127

Figure 6.11 Muon Energy Resolutions For Quasi-elastic And Non-quasi-elastic Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 6.12 Overall Hadronic Energy Resolutions. . . . . . . . . . . . . . . 129

Figure 6.13 Neutrino Energy Resolution. . . . . . . . . . . . . . . . . . . . . 130

Figure 6.14 Neutrino Energy Distributions. . . . . . . . . . . . . . . . . . . 131

Figure 6.15 Quasi-elastic And Non-quasi-elastic True Minus Reconstruct-ed Neutrino Energies. . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 6.16 Quasi-elastic And Non-quasi-elastic Neutrino Energy Reso-lutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 6.17 Comparison Of Neutrino Energy Resolutions. . . . . . . . . . 134

Figure 6.18 Quasi-elastic And Non-quasi-elastic Neutrino Energy Distri-butions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 6.19 Neutrino Types: Fractions And Energy Distributions. . . . . 135

Figure 7.1 Mean Energy Deposition Per Unit Length. . . . . . . . . . . . 139

Figure 7.2 Comparison Of Data And Monte Carlo Simulated LongestTrack Length Distributions. . . . . . . . . . . . . . . . . . . . . 140

Figure 7.3 Energy Estimation Systematic Uncertainty As A FunctionOf The Neutrino Energy. . . . . . . . . . . . . . . . . . . . . . . 140

Figure 7.4 Channels Configurations Systematic Uncertainty As A Func-tion Of The Neutrino Energy. . . . . . . . . . . . . . . . . . . . 142

Figure 7.5 Example Of The ReWeight Package Output. . . . . . . . . . 144

xv

Figure 7.6 GENIE Parameters Systematic Uncertainty As A FunctionOf The Neutrino Energy. . . . . . . . . . . . . . . . . . . . . . . 146

Figure 7.7 GENIE Parameters Total Systematic Uncertainty As A Func-tion Of The Neutrino Energy. . . . . . . . . . . . . . . . . . . . 146

Figure 7.8 Unfolding Algorithm Systematic Uncertainty As A FunctionOf The Neutrino Energy. . . . . . . . . . . . . . . . . . . . . . . 147

Figure 7.9 Summary Of Systematic Uncertainties. . . . . . . . . . . . . . 149

Figure 7.10 Systematic Uncertainty On The Number Of Events As AFunction Of The Neutrino Energy. . . . . . . . . . . . . . . . . 150

Figure 7.11 δN/N As A Function Of The Neutrino Energy. . . . . . . . . 151

Figure 8.1 Angles Of The Longest Track. . . . . . . . . . . . . . . . . . . . 152

Figure 8.2 Longest Track Angular Distributions Without Cosmic Back-ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 8.3 Reconstructed Muon Energy Distributions, No Cosmic Back-ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 8.4 Cumulative Distribution Functions For Muons. . . . . . . . . 157

Figure 8.5 Hadronic Energy Distributions, No Cosmic Background. . . 157

Figure 8.6 Selected Candidate Events Energy Distributions. . . . . . . . 158

Figure 8.7 Efficiency And Purity Matrix For The Quasi-elastic AndNon-quasi-elastic Samples. . . . . . . . . . . . . . . . . . . . . . 160

Figure 8.8 Sample Of Simulated Quasi-elastic Neutrino Events. . . . . . 162

Figure 8.9 Sample Of Simulated Non-quasi-elastic Neutrino Events. . . 164

Figure 8.10 Sample Of Quasi-elastic Neutrino Candidates. . . . . . . . . . 166

Figure 8.11 Sample Of Non-quasi-elastic Neutrino Candidates. . . . . . . 167

Figure 8.12 Number Of Neutrino Candidates vs. Neutrino Energy, NoCosmic Background. . . . . . . . . . . . . . . . . . . . . . . . . . 169

xvi

Figure 8.13 Number Of Neutrino Candidates vs. Neutrino Energy, NoCosmic Background, Shape Comparison. . . . . . . . . . . . . 169

Figure 8.14 Quasi-elastic And Non-quasi-elastic Unfolded Number OfNeutrino Candidates vs. Neutrino Energy, No Cosmic Back-ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Figure 8.15 Quasi-elastic And Non-quasi-elastic Unfolded Number OfNeutrino Candidates vs. Neutrino Energy, No Cosmic Back-ground, Shape Comparison. . . . . . . . . . . . . . . . . . . . . 171

Figure 8.16 Number Of Neutrino Candidates vs. Neutrino Energy, Com-parison Of Monte Carlo Simulation And Data. . . . . . . . . 172

Figure 8.17 Unfolded Number Of Neutrino Candidates vs. Neutrino En-ergy, Comparison Of Monte Carlo Simulation And Data. . . 172

Figure 9.1 Charged Kaon Longitudinal And Transverse Momenta. . . . 177

Figure 9.2 Ratio Of Neutrinos From Charged Kaon Decays Over Neu-trinos From Charged Pion Decays After Event Selection Cri-teria Applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure 9.3 GENIE Cross Sections As A Function Of Neutrino Energy. 183

Figure 9.4 GENIE Uncertainty In The Muon Neutrino Cross Section. 183

Figure 9.5 Reconstruction Efficiency As A Function Of The NeutrinoEnergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Figure 9.6 Total Flux Of Muon Neutrinos Coming From Charged KaonDecays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Figure 9.7 Flux Systematic Uncertainties. . . . . . . . . . . . . . . . . . . 186

Figure 9.8 Data Over Monte Carlo Simulated Flux Ratio. . . . . . . . . 187

Figure 9.9 Total Flux Of Muon Neutrinos Coming From Charged KaonDecays Discriminated By Interaction Type. . . . . . . . . . . 187

Figure 9.10 Quasi-elastic Over Non-quasi-elastic Flux Ratio. . . . . . . . 188

Figure 9.11 Muon Neutrino Monte Carlo Simulated Flux Prediction. . . 191

xvii

Figure 9.12 Reconstruction Efficiency As A Function Of The NeutrinoEnergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Figure 9.13 Inclusive Muon Neutrino Charged Current Cross SectionPer Nucleon On A Carbon Target. . . . . . . . . . . . . . . . . 192

Figure 9.14 Cross Section Systematic Uncertainties. . . . . . . . . . . . . . 193

Figure 9.15 Data Over Monte Carlo Simulated Muon Neutrino ChargedCurrent Cross Section Ratio. . . . . . . . . . . . . . . . . . . . 193

Figure 9.16 Measurements Of Muon Neutrino Charged Current Inclu-sive Scattering Cross Sections. . . . . . . . . . . . . . . . . . . . 194

Figure A.1 Higgs Scalar Potential. . . . . . . . . . . . . . . . . . . . . . . . 200

Figure B.1 Sample Event Illustrating The Tracking Algorithm. . . . . . 205

Figure B.2 Flow Chart Of The Tracking Algorithm. . . . . . . . . . . . . 206

Figure B.3 Sample Event Of The Tracking Algorithm Output. . . . . . . 208

Figure C.1 Neutrino Energy Distributions Before And After Unfolding. 217

Figure C.2 Results From Unfolding Fake Data. . . . . . . . . . . . . . . . 218

xviii

Chapter 1

Introduction

Neutrinos are among the most mysterious of all the known particles. Neutrinos are weakly

interacting particles, which makes them extremely hard to detect, and yet neutrinos partici-

pate in a wide range of phenomena from the subatomic to the cosmological scales. Neutrinos

are key to type-II supernovae core collapse since these carry out of the exploding star about

99% of the gravitational energy release [1]. Neutrinos actively participate in the cooling of

stars that passed the He-burning stage and therefore strongly determine the lifetime of the

star [2]. Their finite mass presents the first evidence of physics beyond the Standard Model,

since the latter assumes that neutrinos are massless. This leads to ideas that would extend

the number of fundamental particles providing a mechanism for neutrinos to acquire their

mass; however through a mechanism not described in the Standard Model. As massive and

extremely abundant particles, neutrinos contribute significantly to the evolution of the Uni-

verse [3]. Massive neutrinos exhibit oscillations that change their identities, and the study

of this phenomenon could provide direction in the solutions of current mysteries, e.g. the

origin of the matter-antimatter asymmetry of the observable Universe.

1

1.1 Brief History Of Neutrinos

In the 1930’s experimental results showed that electrons in beta decays:

n→ p+ e− + energy, (1.1)

were not monochromatic, but rather had a discrete energy spectrum. At that time, the

origins of the beta decay were unknown. However, it was certain that the final products of the

decay were a proton and an electron. Therefore, their energies ought to be fixed. To explain

the undetectable energy, N. Bohr suggested that the concept of conservation of energy could

be discarded. W. Pauli addressed his colleagues as “liebe radioaktive Damen un Herren” [4]

during the 1930 Tubingen meeting to share with them his deep concern about the continuum

energy spectrum of electrons. Pauli suggested that an undetectable particle carried away

the missing energy, a hypothesis that would never be verified, should the particle be truly

undetectable. E. Fermi named Pauli’s hypothetical particle the neutrino1. Over twenty

years later, F. Reins and C. L. Cowan [5] finally discovered the neutrino. The neutrinos2

emitted by a nuclear reactor interacted with protons in two tanks of water, creating neutrons

and positrons. The gamma rays created in the annihilation of the positrons were detected

in the interspersed tanks of liquid scintillator. These neutrinos were later identified as the

electron neutrinos.

Neutrinos are produced in both pion and muon decays. However, it was not clear if the

neutrinos in these two decays were all electron neutrinos, or some belonged to a different

type of neutrinos. In 1962, L. Lederman and J. Steinberger discovered that there existed two

1Italian for little neutral one, to distinguish it from the much heavier neutron.2The reaction is: νe + p→ n+ e+, with antineutrinos.

2

types of neutrinos [6], the electron neutrino (νe) and the muon neutrino (νµ). The 1988 Nobel

Prize in physics was awarded to them for their discovery [7]. At this time, a model of hadrons

and leptons included: the electron, the muon, their corresponding neutrinos, and the up,

down, and strange quarks. The absence of kaon decays to a down quark via a strangeness

changing neutral current, along with symmetry arguments, suggested the existence of a

fourth quark [8], charm. This hypothesis was verified by the discovery of the bound state of

charm-anticharm quarks, the ψ/J meson [9, 10]. By 1977 the existence of the third family

was confirmed with the discovery of the bottom quark at Fermilab, by L. Lederman’s team

[11], and the discovery of the tau lepton, by the SLAC-LBL team [12]. Due to the rarity of

tau neutrinos (ντ ) in neutrino beams, it took over twenty years to experimentally confirm

its existence, by the DONUT collaboration, in 2000 [13].

1.2 Motivations For The NOνA Experiment

Decades of speculation about massive neutrinos3 ended when Super-Kamiokande [16] and

the Sudbury Neutrino Observatory [17] first reported strong evidence of atmospheric and

solar neutrino oscillations, respectively. Neutrino oscillations are only possible if neutrinos

are massive. The Super-Kamiokande result presented evidence that νµ oscillate primarily

into ντ , however they did not achieve a conclusive result on the νµ to νe oscillation. The lack

of evidence in the latter oscillation channel motivated, at that time, the neutrino community

to address the role of the νe in oscillations phenomena. Three experiments: Double Chooz

[18], Daya Bay [19], and RENO [20], presented positive results in 2012 on the νµ → νe

3B. Pontecorvo first suggested, in 1957, the concept of neutrino oscillations [14]. R.Davis found the first experimental indication of neutrino oscillations with his Homestakeexperiment, where a third of the expected flux of neutrinos from the Sun was measured [15].This is known as the solar neutrino problem.

3

oscillation channel showing that the oscillation indeed occurs. This last result presents an

unrivaled opportunity to study the occurrence of CP violation in the neutrino sector [21].

The NuMI Off-Axis νe Appearance (NOνA) experiment, will study the neutrino oscilla-

tions phenomenon. NOνA will investigate the oscillations of νµ to νe, and muon antineu-

trinos (νµ) to electron antineutrinos (νe), using a neutrino beam produced at Fermilab and

detected at the NOνA far detector located near Ash River, Minnesota. NOνA will determine

any differences that occur in the oscillations of neutrinos and antineutrinos when the beam

travels through matter. The neutrino beam at Fermilab is designed to run either in primarily

neutrino or primarily antineutrino modes. Studying the differences between the interactions

of neutrinos and antineutrinos with matter will help to address the following topics:

• The mass ordering of the three neutrino mass eigenstates.

• The value of a CP violating phase in the neutrino sector.

• Smaller uncertainties in the current values of the neutrino oscillation parameters.

The quarks and charged leptons sectors exhibit normal mass orderings, i.e. the member

of the third family is much more massive than the members of the other two families. In

the neutrino sector, however, the possibility of one light and two (quasi-degenerate) heavier

neutrinos could occur4. Even though the absolute value of the mass gap between the lightest

and heaviest neutrinos is known, it is yet to be determined which eigenstate is the lightest

and which is the heaviest. This results in the uncertainty on the sign of the mass difference.

This sign has a profound influence in the νµ → νe oscillation enhancing (+) or suppressing

(-) it. NOνA is going to make a precise measurement of this oscillation channel, which will

determine the sign of the mass difference, and thus the neutrino mass ordering.

4This is know as inverted mass ordering or hierarchy. See figure 2.9.

4

By taking neutrino and antineutrino measurements, NOνA will be able to establish

whether there is any significant difference in the two oscillation patterns beyond that intro-

duced by propagation of neutrinos through matter, indicating that CP symmetry is violated

in neutrino oscillations. Therefore, neutrinos become prime candidates to explain the origin

of the observed matter-antimatter asymmetry at the time of the Big Bang.

NOνA is a two-detector experiment with one located underground at Fermilab (near

detector), and the other, with a mass of 14 kton, on the surface in northern Minnesota

(far detector). The two detectors are constructed from the same materials differing only in

mass (300 ton for the near detector), and use the same readout electronics differing only in

sampling rate (higher for the near detector). The similarity between the detectors allows

the initial event rate of νe and νµ, measured by the near detector, to yield a nearly bias-

free normalization of the event rate at the far detector. The two detectors are 14 mrad

off the neutrino beam axis, which provides a narrow neutrino energy spectrum5 peaked

around 2 GeV. At the far detector, a 2 GeV neutrino beam combined with its baseline

length of 810 km produces a maximum in the νe appearance probability, and a minimum in

the νµ survival probability, with the flux dominated by oscillations to ντ . The exclusively

neutral current interactions of ντ at this energy, where the scattered ντ carries away a large

fraction of the energy, leaves little hadronic energy to produce backgrounds for the 2 GeV

νe interactions.

The two NOνA detectors are matrices of PVC tubes filled with liquid scintillator, re-

sulting in detectors with approximately 70% of their weight being active detecting material.

The ionization left by charged particles created in the neutrino interactions generates light

in the scintillator, which is collected by a special kind of fiber optics that transport the

5See figure 3.3.

5

wavelength-shifted light to avalanche photodiodes. The photodiode electrical pulses are am-

plified, digitized, and processed by a data acquisition system. The processed data are stored

on disc for later analysis.

The NOνA collaboration built a prototype detector at Fermilab exposed to the neutrino

beam to develop calibration, simulation, and reconstruction tools, using real data. A number

of construction issues were identified and solved. These influenced the final design of the

near and far detectors. Regardless of the location of the prototype on the surface, the

neutrino interaction signals can be separated from the enormous amounts of cosmic rays

that continuously illuminate the prototype detector. Cosmic rays are a background to the

νµ charged current neutrino signal6. The neutrino beam is pulsed at regular intervals, and the

analysis of the data finds a peak at the time of the interaction candidates from the neutrino

beam. With the ability to distinguish neutrino events from cosmic rays, the prototype is

used to make a proof of principle: the collaboration searched for νµ interactions and these

were indeed found [22].

The νµ charged current quasi-elastic cross section7 was measured [22] in the NOνA

prototype detector using a selection of events with one and only one reconstructed track.

The neutrino energy was determined solely from the range and angle of the reconstructed

muon track. Using the calorimetric capabilities of the prototype detector’s design, and

the neutrino flux embedded in the NOνA Monte Carlo simulation, this thesis presents an

estimate of the inclusive νµ charged current cross section.

Charged kaons produced at the NuMI target8 are the source of the νe that contribute

6See section 2.1.1 for a definition of the charged current interaction.7See section 2.1.1.1 for a definition of the charged current quasi-elastic interaction.8See section 3.1 for a detailed description of the NuMI beam, including the charged kaon

production.

6

to the total background of the νµ beam that reaches the far detector. Knowing the initial

content of νe in the neutrino beam will minimize the systematic uncertainty in the number

of νe expected from νµ oscillations. The charged kaon decay peak in the neutrino energy

is observed in the inclusive charge current interaction data to be presented. The off-axis

angle, and the two-body decay of the mesons, lead to a narrow peak in the energy of the

neutrinos at 2 GeV. However, the production rate of charged kaons in the NuMI target has

to be determined from Monte Carlo simulations as there are no direct measurements. At the

location of the prototype detector, the flux of 2 GeV neutrinos is nearly all from the decay

of charged kaons. The last two statements lead to a rather large normalization uncertainty

in the measured neutrino cross section. From the few other measurements of neutrino cross

sections in this energy regime, the level of uncertainty in the flux can be estimated. Therefore,

both interpretations of the data, as an inclusive cross section estimate, or as a flux estimate,

are presented.

1.3 Outline

In order to set the theoretical framework behind the neutrino interactions, chapter 2 presents

the basics of the weak interactions, emphasizing those involving neutrinos (section 2.1). In

addition, the possible origins of neutrino masses (section 2.2), and a review of the formalism

of neutrino oscillations (section 2.3) are presented.

Chapter 3 provides an overview of the Fermilab neutrino beam that supplies the neutrinos

to the NOνA experiment. Sections 3.1 and 3.2 summarize the components of the neutrino

beam, and the various steps involved in the neutrino production. The beam composition

and energy spectrum are presented in section 3.3.

7

The concepts behind the NOνA experiment are presented in chapter 4. Section 4.1 is an

overview of the experiment. The prototype detector is introduced in section 4.2. The various

software tools used for data processing and analysis are presented in section 4.3. The data

quality checks are reviewed in section 4.4. A brief study of cosmic rays, and its implications

in the experiment, is provided in section 4.5.

The νµ event selection procedure is described in chapter 5. The differences between

events from charged and neutral current interactions are discussed in section 5.1, the event

containment criteria are presented in section 5.2, and the charged current event selection is

explained in section 5.3.

The neutrino energy reconstruction process is discussed in chapter 6. There are two

steps in this process: muon energy reconstruction (section 6.1), and the hadronic energy

reconstruction (section 6.2). A classification between quasi-elastic and non-quasi-elastic

events (section 6.3) is necessary to better reconstruct the neutrino energy (section 6.4).

The various systematic uncertainties associated with the results are detailed in chapter

7. The analysis of the data is presented in chapter 8. A comparison between simulation

and data is provided in section 8.1, and the results are presented in section 8.2. Chapter 9

discusses the implications of the results on: the neutrino flux (section 9.1), and the inclusive

νµ charged current cross section (section 9.2). Final remarks are presented in section 9.3.

8

Chapter 2

Theoretical Framework for Neutrinos

The Standard Model provides the theoretical foundations for quantum chromodynamics

(QCD), and it also unifies electromagnetism and the weak interaction, based on the works

done by S. L. Glashow [23], S. Weinberg [24], and A. Salam [25]. Neutrinos have an important

role in the weak interaction, which is briefly reviewed in this chapter. Massive neutrinos are

not part of the Standard Model of particle physics. There are a few ideas to explain the

origins of neutrino masses, the most popular within the neutrino community is the See Saw

Mechanism, also reviewed in this chapter. Neutrino oscillations are one of the most relevant

consequences of massive neutrinos, and therefore are reviewed at the end of the chapter.

2.1 Neutrinos In Weak Interactions

The electroweak interaction was first developed in a phenomenological fashion, and later

complemented when additional measurements became available. In 1934 Fermi [26] intro-

duced an effective Hamiltonian1:

Heff = −GF√2Jµ(x)J

†µ(x), (2.1)

1Fermi coupling constant: GF /(hc)3 = 1.166 × 10−5 GeV−2.

9

to model the low energy charged current weak interaction of hadrons and leptons known at

the time as:

Jµ(x) = Ψνeγµ(1 − γ5)Ψe + Ψpγµ(1 − γ5)Ψn. (2.2)

The model resembled that of electromagnetism given its vector part, however it also included

an axial part introduced after the discovery of parity violation [27].

At low neutrino energies, charged current interactions are characterized by quasi-elastic

scattering with a nucleon and lepton in the final state. Nucleon resonance production be-

comes important as their thresholds are crossed. At high energies, the cross section grows

linearly with energy and becomes dominated by deep inelastic scattering. The formalism

describing these interactions is described in the next section.

2.1.1 Charged Current Interactions

The interaction of the W boson with fermions has the following Lagrangian:

LW = − g

2√

2

(

JµWW−

µ + Jµ†WW+

µ

)

, (2.3)

where the charged current (CC) is:

Jµ†W =

3∑

k=1

(

νkγµ(

1 − γ5)

lk + ukγµ(

1 − γ5)

dk)

,

= (νe νµ ντ )γµ(

1 − γ5)

Ul

e

µ

τ

+ (u c t)γµ(

1 − γ5)

Vq

d

s

b

. (2.4)

10

Vq is the Cabibbo-Kobayashi-Maskawa (CKM) matrix [28, 29], and Ul is the Pontecorvo-

Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix [30, 31]. The vertices of interest,

from equations (2.3, 2.4), are: djuiW−, uidjW

+, ljνiW−, and νiljW

+, which contribute to

the neutrino scattering with nucleons. The first two vertices correspond to the lower vertex

in figure 2.1, and the last two correspond to the upper vertex in figure 2.1. The interaction:

νµ + d → µ− + u, which is the underlying process of the νµ + n → µ− + p scattering, is an

example of the interactions that are generated with the four vertices mentioned above.

Figure 2.1 Weak Interaction Charged Current Representation Of Neutrino-nucleon Scattering. In the drawing the index l represents all charged leptons, the letterd represents all the Q = −e/3 quarks, and the letter u represents all the Q = 2e/3 quarks.

2.1.1.1 Quasi-elastic Scattering

For energies smaller than the W mass, the amplitude (M):

M =GF√

2lγµ(1 − γ5)ν〈p|J+

µ |n〉, (2.5)

11

of the neutrino interactions2:

νl(k) + n(p) → l−(k′) + p(p′),

νl(k) + p(p) → l+(k′) + n(p′), (2.6)

is the product of the hadronic and leptonic currents [27]:

The general hadronic current gets contributions from the motion of the quarks within

the hadrons, hence the introduction of form factors is useful to describe the physics of the

processes3:

〈p|J+µ |n〉 = n

(

gV γµ + fVpµ

2MN+ hV

qµ

2MN+ gAγµγ5 + ıfAσµν

qνγ5

2MN+ hA

qµγ5

2MN

)

p. (2.7)

Here the gA, fA, and hA are axial form factors, and the gV , fV , and hV are vector form

factors. The product hAqµγ5 is proportional to the lepton’s mass, therefore is a negligible

term4, and hV and fA need to be zero due to charge conjugation and time-reversal arguments

[27]. The simplified form of equation (2.7) is:

〈p|J+µ |n〉 = n

(

gV γµ + fVpµ

2MN+ gAγµγ5

)

p. (2.8)

The kinematics for the processes (see equation (2.6)) in the laboratory frame, for which the

2k and p are momenta.3MN is the mass of the target nucleon.4True for light leptons.

12

nucleon is at rest, are given by:

p · k = p′ · k′ = MNEν ,

p · k′ = p′ · k = MNEν +q2

2,

k · k′ =m2

l

2− q2

2, (2.9)

p · p′ =M2

W

2− q2

2,

Q2 = 4EνE′ sin2

(

θ

2

)

,

where Eν is the energy of the incoming νl, E’ is the energy of the outgoing l, Q2 is the

momentum transfer, and θ is the angle between the momenta of the incoming neutrino and

the outgoing charged lepton. Equations (2.8, 2.9) lead to the differential cross section:

dσQE

dE′ =G2

FMN

4π

[

(gV − gA)2 + (gV + gA)2(

E′

Eν

)2

+ (g2V − g2A)MN

E2ν

]

+G2

FMN

8π

[

f 2V η + 2fV gV

]

[

(

1 +E′

Eν

)2

− Q2

E2νη

]

, (2.10)

with:

η = 1 +Q2

4M2N

. (2.11)

Monte Carlo simulated quasi-elastic cross sections based on this function provide a rea-

sonable representation of the data from experiments (MiniBooNE [32], ANL [33], BEBC [34],

BNL [35, 36], FNAL [37], Gargamelle [38, 39], LSND [40], NOMAD [41], Serpukhov [42],

and SKAT [43]), as seen in figure 2.2. The two solid curves are the Monte Carlo simulation

predictions of the cross sections based on equation (2.10). The quasi-elastic cross sections

become constant for energies above 10 GeV.

13

Figure 2.2 Muon Neutrino Quasi-elastic Cross Sections. Measurements of the νµ(black) and νµ (red) quasi-elastic scattering cross sections (per nucleon) as a function ofneutrino energy [44].

2.1.1.2 Resonant Pion Production

As neutrino energy gets larger, their scattering with nucleons can excite the latter to higher

energy levels; these interactions produce baryon resonances (N∗):

νµN → µ− +N∗, (2.12)

which rapidly decays (often) into a nucleon and a charged pion:

N∗ → π +N ′, (2.13)

where N and N ′ could be either a neutron or a proton in a nucleus. Higher order multiplicity

decay modes are also possible. These resonances were studied by D. Rein and L. M. Sehgal

14

(a) (b)

Figure 2.3 Muon Neutrino Resonance Cross Sections. (a) Resonance production crosssection for differing nH [47]. (b) Existing measurements of the cross section for the processin equation (2.15) as a function of energy [48].

[45], based on the model of baryon resonances of R. Feynman, M. Kislinger, and F. Ravndal

[46]. Their work describes the resonances in terms of excited states of three bounded quarks

(a, b, and c) with a relativistic harmonic oscillator potential:

Hres = 3(p2a + p2

b + p2c) +

1

36Ω2[(ua − ub)

2 + (ub − uc)2 + (uc − ua)2], (2.14)

with Ω2 = m2ω20, the u’s are conjugate positions of the quarks, and the p’s are four-

momentum vectors of the quarks5. The eigenvalues of the harmonic oscillator presented

in [46] are a succession of integers, nH , times Ω. The first state, nH = 0, is the reso-

nance ∆(1232), and its cross section saturates at energies around 2 GeV. In the cases where

1 ≤ nH ≤ 2, the theoretical cross sections for: nH = 1 saturates at energies higher than

20 GeV, and nH = 2 rises linearly with energy, as seen in figure 2.3a. Measurements of the

cross section for the process:

νµ + p→ µ− + p+ π+, (2.15)

5u’s and p’s defined in [46].

15

have been made in bubble chamber experiments, and the results are shown in figure 2.3b.

2.1.1.3 Deep Inelastic Scattering

When hadron structures are examined on a very short distance scale, it is found that these

are a set of non-interacting quarks. Short distance, in this case, translates to high energy

electroweak interactions in deep inelastic (lepton-hadron) scattering of the form:

l +N → l′ + X, (2.16)

where X includes single nucleon or nucleon resonance production. The kinematics of these

processes are described in terms of the following variables (recall equations (2.9)):

x =Q2

2MNEy, y =

Ehad

Eν, (2.17)

where Ehad is the energy of the hadronic system. Using these variables, the deep inelastic

scattering (DIS) double differential cross section is:

d2σ

dxdy=

G2FMNEν

π

(

1 + Q2

M2W

)2

[

2xy2

2(gV + fV ) − fV

(

1 − y − MN xy

2Eν

)]

. (2.18)

Equation (2.18) is the core of the physics involved in DIS. However, additional effects must be

included in any realistic description. The inclusion of non-perturbative higher twist effects

[49], heavy quark production [50], target mass effects [51], nuclear effects and radiative

corrections [52], higher order QCD processes [53], and lepton masses [54], further modify the

scattering cross section in equation (2.18). In general, these contributions are known with

16

reasonable uncertainties [48].

(a) (b)

Figure 2.4 Total Inclusive Muon Neutrino Charged Current Cross Sections. Total(inclusive) (a) νµ and (b) νµ charged current cross sections per nucleon over neutrino energyas a function of neutrino energy. The total cross sections (solid lines) are the sum of: QE(dashed), resonance (dot-dashed), and deep inelastic (dotted) [48].

A collection of data acquired over the last 50 years, using different experimental tech-

niques [48], is presented in figure 2.4. These figures show the measured and predicted QE

and inclusive CC cross sections. For energies higher than 3 GeV the inclusive cross section is

dominated by the DIS processes, as can be seen in the region where the ratio σDIS/σQE > 1.

To isolate the DIS events, neutrino experiments usually apply kinematic cuts to remove QE

scattering and resonance contributions from the data.

2.1.1.4 Inclusive Cross Section Measurements

There are only two experiments that have measured the νµ CC inclusive cross section at

neutrino energies around 2 GeV: Brookhaven’s 7 ft bubble chamber [55] (1982) and SciBoone

(2011). The experiment at Brookhaven used a 7 ft deuterium bubble chamber exposed to the

Alternating Gradient Synchrotron wide-band neutrino beam. The value of the inclusive cross

section divided by the neutrino energy falls significantly from 0.4 GeV to 1.5 GeV, and then

17

becomes constant, as seen in figure 2.5. Above 1.6 GeV, the measured inclusive CC cross

section per nucleon divided by the neutrino energy is σ/E = (0.80±0.03)×10−38× cm2/GeV

[55], the average of the values shown in figure 2.5.

Figure 2.5 Brookhaven National Laboratory Muon Neutrino Inclusive Cross Sec-tions. νµ inclusive cross section over neutrino energy as a function of the neutrino energyfor: ν p and ν n interactions. The solid lines are the best fits for neutrino energies higherthan 1.6 GeV [55].

At neutrino energies around 2 GeV, nuclear effects are important. When the W boson’s

four-momentum is low enough such that its de Broglie wavelength is comparable to the

size of the target nucleus, the scattering involves the entire nucleus. The cross sections on

deuterium targets are not directly applicable to the heavier nuclear target materials used in

current accelerator-based neutrino experiments. In neutrino scattering on a nucleon (in a

nuclear target), if there is a small momentum transfer to the final state nucleon, this can

only exist in excited states allowed by Pauli’s Exclusion Principle. In the Fermi gas model,

the excited nucleons undergo transitions across the Fermi surface, with momentum around

18

Figure 2.6 SciBooNE Muon Neutrino Inclusive Cross Sections. νµ inclusive crosssection per nucleon on a polystyrene target (C8H8) [56]. The NEUT (dashed red line) [57]and NUANCE (black line) [58] predictions are shown for reference.

250 MeV/c, from low energy states, which are all occupied, to high energy states. Thus, the

neutrino-nucleon interaction can only occur if there is an available higher energy state for

the nucleon to occupy. This is known as Pauli-Blocking [59]. When the nucleon involved

in the scattering interacts with other nucleons, the kinematics and multiplicity of the final

state are diverse. SciBoone measured the inclusive νµ CC interactions, using the Fermilab

Booster neutrino beam, on a polystyrene target. Their results are shown in figure 2.6. The

rise of the cross section with energy agrees somewhat better with the predictions of the

NEUT simulation. The smaller error bars show the uncertainties of the rate normalization

factors and the larger error bars represent the total error including the flux uncertainties.

The two results presented in this section are consistent with each other, and are useful

to tune various neutrino interaction models in the 2 GeV region.

19

2.1.2 Neutral Current Interactions

Though hints for the existence of neutrino scattering without final state leptons had been

around for a number of years, compelling experimental evidence for weak neutral current

interactions [60] became available at about the same time as the electroweak unification

theory was proposed, and it depended crucially on the existence of weak neutral currents.

The fermionic Lagrangian6 describes neutral and charged currents. There is a La-

grangian, equivalent to that in equation (2.3), that describes the interaction of the Z boson

with fermions, given by [27]:

LZ =GF√

2

3∑

k=1

νkγµ(

1 − γ5)

νk

ukγµ[

ukL

(

1 − γ5)

+ ukR(1 + γ5)

]

uk

+GF√

2

3∑

k=1

νkγµ(

1 − γ5)

νk

dkγµ[

dkL

(

1 − γ5)

+ dkR(1 + γ5)

]

dk

. (2.19)

The vertices of interest that come out from equation (2.19) are: ujuiZ, djdiZ, and νjνiZ,

which contribute to the neutrino scattering with nucleons. The first two vertices correspond

to the lower vertex in figure 2.7, and the last vertex corresponds to the upper vertex in figure

2.7. The interaction νµ + d → νµ + d is an example of the scattering that is generated

with the three vertices mentioned above.

Equations (2.9, 2.19) lead to the elastic neutral current differential cross section [48]:

dσ

dQ2=G2

FM2N

8πE2ν

[

A± 4MNEν −Q2

M2N

B +

(

4MNEν −Q2)2

M4N

C

]

, (2.20)

6See equation (A.12).

20

Figure 2.7 Weak Interaction Neutral Current Representation Of Neutrino-nucleonScattering. In the drawing the index l represents all charged leptons, the letter q representsall quarks.

with:

A =Q2

M2N

[

g2Aη+ − (gV + fV )2η− + f 2V

Q2

4M2N

η− + (gV − fV )fVQ2

M2N

]

,

B =Q2

M2N

gAfV ,

C =1

4

[

g2A + (gV + fV )2 + f 2V (η− − 1)

]

,

η± = 1 ± Q2

4M2N

. (2.21)

The MiniBooNE collaboration has published a measurement of the elastic neutral current

differential cross section for the process:

ν +N → ν +N, (2.22)

21

on a CH2 target [61], as seen in figure 2.8.

Figure 2.8 MiniBooNE Flux-averaged Neutral Current Differential Cross Section.The blue line is the predicted spectrum of elastic neutral current background which has beensubtracted out from the total differential cross section [61].

C. H. Llewellyn-Smith [62] showed that the semileptonic neutral and charged currents7

cross sections are related8 through sin θW by:

Rν =σNC

σCC=

[

1

2− sin2 θW +

5

9sin4 θW

]

(1 + r), (2.23)

where r is the ratio of the antineutrino charge current cross section to the neutrino one. The

CHARM collaboration [63] measured this ratio for energies larger than 2 GeV to be:

Rν = 0.317 ± 0.006. (2.24)

7Charged current QE interactions.8See equation (A.22).

22

2.2 Neutrino Masses

The formalism9 by which particles acquire their mass in the Standard Model requires both

left-handed10 and right-handed11 states for a given particle to have a mass term in the

Lagrangian. Neutrinos are a special case since there is no evidence that supports the existence

of right-handed neutrinos. Yet, there is no argument or symmetry within the Standard Model

that explicitly forbids the existence of such neutrinos. In the case of the photon12, the gauge

symmetry requires it to be massless.

In view of this, there are two possible mass terms: the Dirac term13:

mD(ψRψL + h.c.), (2.25)

and the Majorana term:

mM (ψcLψL + h.c.), (2.26)

where ψc = Cγ0ψ∗ is the charge conjugated field of ψ, and ψcL ≡ (1+γ5)ψ

c/2 is right-handed.

The Majorana term in equation (2.26) violates lepton number conservation by two units14

and makes neutrinos indistinguishable from antineutrinos [64]. All the massive fundamental

particles in the Standard Model have some type of charge (even though gluons are massless,

these are linear combinations of color states), which changes sign under charge conjugation

9See appendix A for a brief summary of the formalism.10See equation (A.2).11See equation (A.3).12See equation (A.19).13See equation (A.20).14In the Dirac mass term, the matter field has lepton number: +1, and the antimatter

field has lepton number: -1. In the Majorana mass term, both fields have lepton number:+1.

23

providing a clear distinction between particles and antiparticles. A Majorana mass term is

therefore only possible for neutral particles, exhibiting no charges of any kind; a feature that

only neutrinos exhibit.

2.2.1 Seesaw Mechanism

The seesaw mechanism [65] is the simplest renormalizable model that introduces a mass term

for neutrinos in the Lagrangian:

LM = mDψRψL +mM

2ψc

RψR + h.c..

=1

2(ψc

L ψR)

0 mD

mD mM

(

ψL

ψcR

)

+ h.c.. (2.27)

The Lagrangian in equation (2.27) is simplified by diagonalizing its mass matrix. The diag-

onalization procedure defines a new two component field νM , written as:

νM ≡ ψL + ψcL ≡

(

νsmνnsm

)

, (2.28)

such that:

LM =1

2(νsm νnsm)

msm 0

0 mnsm

(

νsmνnsm

)

. (2.29)

The mD scale is in the order of a typical Standard Model fermion, and mM ≫ mD. The

masses of the νiM are [66]:

msm∼=

m2D

mM, mnsm

∼= mM . (2.30)

24

The νsm states are associated with the neutrinos of the Standard Model15, and the νnsm

states represent very massive neutrinos yet to be observed. Equation (2.30) suggests that

the msm are very small, and the model suggests that the mnsm are about 1015 GeV/c2

[66]. The direct laboratory limits on the msm establish an upper limit in the range: (0.3-

0.9) eV /c2, depending on the nuclear model considered [67]. The most relevant feature of the

seesaw mechanism is that it explains the lightness of the known neutrinos, the νsm states,

by setting a very high mass to the yet to-be-observed heavy neutrinos, the νnsm states.

The out-of-thermal-equilibrium decays of these heavy neutrinos are key to the attempts for

explaining the baryon-antibaryon asymmetry of the Universe in terms of leptogenesis [66].

The leptonic part of the Lagrangian can be written in the mass basis of the charged

leptons and of the singlet fermions as follows16 [68]:

Lss = h∗ (Lφc∗) lR − λ∗ (Lφ∗

)

νnsm − 1

2mnsmνnsmν

cnsm + h.c.. (2.31)

with:

(

Lφc∗) = (νL lL)

0 −1

1 0

(

φ−

φ∗0

)

, (2.32)

and where h and λ are matrices of Yukawa couplings. From the second Yukawa term in

equation (2.31), the lightest right-handed heavy neutrino (ν1nsm) can decay into:

ν1nsm → φlL. (2.33)

In the early Universe, when the temperature (TU ) was TU ∼ mM , the ν1nsm population was

15There are three neutrino eigenstates associated with each of the two states in equation(2.28).

16See equations (A.1, A.2, and A.3).

25

stable. Once TU < mM the population of ν1nsm is no longer stable, since they can not be

produced any more, and the ν1nsm all decayed. If there is an asymmetry17 (A) in the decay of

the ν1nsm that favors the production of charged leptons over charged antileptons, there would

be an excess of charged leptons in the early Universe. After charged leptons annihilation,

such excess would be the origin of the matter-dominated Universe seen nowadays. A is

defined in terms of the decay rates as follows [68]:

Akk ≡ Γ(ν1nsm → φlkL) − Γ(ν1

nsm → φlkL)

Γ(ν1nsm → φlL) + Γ(ν1

nsm → φlL), (2.34)

where k denotes the various lepton flavors. By definition: |Akk| ≤ 1. Usually, it is

much smaller than 1. To account for the observed baryon asymmetry, it is required that:

|Akk| > 10−7 [68].

2.3 Neutrino Oscillations

The charged current, given by equation (2.4), introduced the concept of quark and lepton

mixing. The leptonic part of the Lagrangian in equation (2.3) is:

LlW =

g

2W+

µ

(

νeγµe+ νµγ

µµ+ ντγµτ)

+ h.c.. (2.35)

Taking a basis in which the charged leptons’ mass term is diagonal, the neutrino flavor

eigenstates are a linear combination of the neutrino mass eigenstates:

ναL = Uαiν

′iL, να

R = Tαiν′iR, (2.36)

17If A > 0.

26

which leads to the mass term:

LM = ν′iLT†iαmαβUβjν

′iL + h.c., (2.37)

with:

T †mU = mdiag, (2.38)

where U is the PMNS matrix [69] from equation (2.4).

Since neutrinos are only observed by their role in the weak interaction, all observed neu-

trinos are represented in the flavor basis; however, only mass eigenstates propagate through

space and time. If the resolution of real experiments allowed the direct measurement of

neutrino masses in individual processes, the oscillations would not be of scientific interest at

all, i.e. once a neutrino is produced in a mass eigenstate it remains in that mass eigenstate

forever. The current understanding of how neutrinos interact is through the weak interaction

and its flavor eigenstates, thus neutrino oscillations are unavoidable, under these conditions.

The weak interaction produces neutrinos in a given flavor eigenstate, and these are detected

by means of a weak interaction process which involves a given flavor eigenstate. The question

is: would those two states be the same?

In vacuum, a neutrino state of the generation α, after a time interval t, is given by [64]:

|να〉t =∑

i

Uαie−ıEit|ν′i〉, (2.39)

and the transition amplitude to the state νβ is:

〈νβ |να〉t =∑

UαiU†iβe

−ıEit. (2.40)

27

The lightness of neutrinos yields |p| ≫ mi, which means: Ei =√

p2 +m2i ≃ p +m2

i /2E.

With this approximation, and using equation (2.36), equation (2.39) becomes:

|να〉t ≃ e−ıptU

e−ım21t/2E

e−ım22t/2E

e−ım23t/2E

U†|νβ〉,

= e−ıptU

1 − ım2

1t

2E + · · ·

1 − ım2

2t

2E + · · ·

1 − ım2

3t

2E + · · ·

U†|νβ〉. (2.41)

Using equation (2.38) as:

U†m†mU = m2diag =

m21 0

m22

0 m23

, (2.42)

equation (2.41) becomes:

|να〉t ≃ e−ıpt[e−ım

†m2E t

]αβ |νβ〉. (2.43)

Since NOνA will study the oscillation of νµ to νe, it is interesting to see the mixing between

two generations18, which is a valid approximation in this case. The relevant mixing matrix

(U) is [64]:

U =

cos θ sin θ

− sin θ cos θ

. (2.44)

18For the three generation mixing see e.g. [66].

28

Using equation (2.44), m†m can be written as:

m†m = Um2diagU

† =m2

1 +m22

2+

∆m2

2

− cos 2θ sin 2θ

sin θ cos 2θ

, (2.45)

where ∆m2 = m22 − m2

1. Equation (2.45) allows to rewrite equation (2.43) as:

|να〉(t) =

cos ∆m2

4E t− ı sin ∆m2

4E t cos 2θ −ı sin ∆m2

4E t sin 2θ

−ı sin ∆m2

4E t sin 2θ cos ∆m2

4E t+ ı sin ∆m2

4E t cos 2θ

|νβ〉. (2.46)

The transition from |νµ〉 to |νe〉(t) is then given by:

〈νµ|νe〉(t)

= (0 1)

cos ∆m2

4E t− ı sin ∆m2

4E t cos 2θ −ı sin ∆m2

4E t sin 2θ

−ı sin ∆m2

4E t sin 2θ cos ∆m2

4E t+ ı sin ∆m2

4E t cos 2θ

0

1

,

= −ı sin ∆m2

4Et sin 2θ. (2.47)

The physical meaning of the transition: 〈νµ|νe〉(t), is understood through the transition

probability (Pνµ→νe) expressed as:

Pνµ→νe = |〈νµ|νe〉(t)|2 = sin2 ∆m2

4Et sin2 2θ. (2.48)

Since neutrinos travel at nearly the speed of light, to a good approximation: t ≈ L/c, where

L is the distance covered by the neutrino between the creation and detection points. With

this small change, oscillation experiments might place their detectors at distances from the

29

source that fulfill19:

π ≤ ∆m2L

2E. (2.49)

The transition probability is maximal when:

π =∆m2L

2E. (2.50)

The ratio L/E is of paramount importance in neutrino oscillation experiments since it deter-

mines much of the logistics of their experimental setup. All neutrino oscillation experiments

that use a neutrino beam, send it through matter. Interactions with matter introduce changes

in the formalism described in this section since νe interacts with electrons via charged and

neutral current interactions, while νµ and ντ only interact with electrons via the neutral

current interaction. As a consequence, there is a coherent effect in the transitions. See [64]

for a formalism that describes neutrino oscillations in matter.

The formalism presented above is a simplification of the three neutrino oscillations. Nev-

ertheless, this simplification is sufficiently good to allow the full problem to be represented

as a collection of two neutrino transitions for which U is:

U =

1 0 0

0 c23 s23

0 −s23 c23

c13 0 s13e−ıδ

0 1 0

−s13eıδ 0 c13

c12 s12 0

−s12 c12 0

0 0 1

eıϕ12 0 0

0 eıϕ22 0

0 0 1

,

=

c12c13eıϕ12 s12c13e

ıϕ22 s13e

−ıδ

(−s12c23 − c12s23s13eıδ)eı

ϕ12 (c12c23 − s12s23s13e

ıδ)eıϕ22 s23c13

(s12s23 − c12c23s13eıδ)eı

ϕ12 (−c12s23 − s12c23s13e

ıδ)eıϕ22 c23c13

. (2.51)

19In natural units: c = 1 = h.

30

Here: cij ≡ cos θij , and sij ≡ sin θij . θ12, θ23, and θ13, are the mixing angles. The phase

δ is the leptonic analogue of the single phase that appears in the quark mixing matrix, and

ϕ1, and ϕ2 are Majorana phases that are relevant if neutrinos are their own antiparticles.

The importance of the phases is evident when the Lagrangian in equation (2.35) is CP

transformed:

(CP )LlW (CP )−1 =

g

2W−

µ llγµU∗ν +

g

2W+

µ νlγµUT l. (2.52)

If U∗ 6= U , i.e. if U is complex due to δ 6= 0, the weak interaction is not CP invariant in

the neutrino sector. The existence of phases is of great relevance to leptogenesis since these

determine20 whether there is an asymmetry in the production of leptons and antileptons in

the early Universe [66].

In the three neutrino oscillations scenario, the parameter θ13 has an important role in

the transition probability (Pνµ→νe) presented in equation (2.48), which is the probability in

vacuum for the two neutrino case. Once the matter effects are taken into account, equation

(2.48) becomes21:

Pmatterνµ→νe ≈

(

1 ± 2E

ER

)

P vacuumνµ→νe ,

≈(

1 ± 2E

ER

)

∣

∣

∣2U∗

µ3Ue3C1 + 2U∗µ2Ue2C2

∣

∣

∣

2, (2.53)

where ER is the matter resonance energy associated with the atmospheric mass difference

(∆m232) and the electron number density in Earth (Ne):

ER =∆m2

32

2√

2GFNe. (2.54)

20The decay of right-handed neutrinos in the seesaw mechanism provides the asymmetry.21The ± in equation (2.53) is: + for neutrinos and - for antineutrinos.

31

Parameter Value

sin2 θ13 0.0219+0.0010−0.0011

sin2 θ23 0.451 ± 0.001 ⊕ 0.557+0.027−0.035

sin2 θ12 0.304 ± 0.012

∆m231 (2.458 ± 0.002) × 10−3 eV2

∆m232 (−2.488 ± 0.047) × 10−3 eV2

∆m221

(

7.50+0.19−0.17

)

× 10−5 eV2

Table 2.1 Neutrino Oscillation Parameters. Best measurements of the neutrino oscilla-tion parameters [70].

C1 and C2 are oscillation parameters not relevant at the moment. From the transition matrix

in equation (2.51), P vacuumνµ→νe depends on:

2U∗µ3Ue3 = e−ıδ sin 2θ13 sin θ23. (2.55)

The experimental results obtained by Double Chooz, Daya Bay, and RENO measuring a

non-zero θ13 allow the possibility for CP violation in the neutrino sector since the transition

probability presented in equation (2.53) depends on the phase δ. The transition probability

P vacuumνµ→νe changes the sign of δ, which directly affects the measurement of the asymmetry

(A):

A =

∣

∣

∣

∣

P − P

P + P

∣

∣

∣

∣

. (2.56)

If it is non-zero, δ is non-zero, resulting in the mentioned CP violation. Both A and Akk

(recall equation (2.34)) are influenced by the phases δ and ϕ. A CP violation in the light

neutrino sector could have CP violation implications in the heavy neutrino sector, which is

the key to leptogenesis, as mentioned in section 2.2.1.

All the free parameters in the neutrino oscillation formalism have been measured [70],

except for the phases. Table 2.1 summarizes the most accurate measurements of these

32

parameters. Two open questions concerning these parameters remain unanswered, and are

the subject of study of the next generation of neutrino oscillation experiments, including

NOνA. The first is related to the octant of θ23, since this is measured through the expression

sin2 2θ23, its measured value only shows that it is close to π/4. However, whether θ23 is

smaller or greater than π/4 remains uncertain. The possibility for θ23 = π/4 implies that ν3

would have exactly the same amount of νµ and ντ . The second question concerns the mass

ordering, or mass hierarchy, of the mass eigenstates. Oscillation measurements are related

to sin2 ∆m2/4E (see equation (2.48)), where the sign of the ∆m2 is undetermined. The

current knowledge is that there are two quasi-degenerate mass eigenstates, m1 and m2, and

a third one, m3, that is either much heavier (normal hierarchy), or much lighter (inverted

hierarchy) than the other two, as shown in figure 2.9. In the normal hierarchy scenario,

neutrino oscillations are enhanced and the antineutrino oscillations are attenuated, for the

inverted hierarchy the opposite is true.

Figure 2.9 Neutrino Mass Hierarchy.

33

Chapter 3

NuMI Beam Overview

Figure 3.1 Fermilab Accelerator Campus. Picture from Google Maps.

Figure 3.2 NuMI Beam Trajectory. Taken from [71].

The accelerator complex at Fermilab is currently dedicated to providing protons from

the Main Injector (MI) to seed the neutrino beam that is used by the NOνA experiment for

34

their neutrino oscillation investigations, and used by other experiments that study neutrino

interactions. In the next few years, accelerator-based neutrino experiments may resolve the

neutrino mass hierarchy, the octant ambiguity in θ23, and begin to see hints that there is a

large violation of the CP symmetry in neutrino oscillations.

Figure 3.3 NuMI Beam Energy Spectrum. Neutrino event rates as a function of neutrinoenergy and off-axis angle [72].

The MI [73] is a very high intensity1 proton accelerator located at Fermilab (see figure

3.1). The MI provides 120 GeV protons that serve as input to the Neutrinos at the Main

Injector (NuMI) beam, which has a power2 of 320 kW. This beam travels from the Fermilab

Accelerator Campus, through the Earth, to the MI Neutrino Oscillation Search (MINOS)

underground laboratory at Soudan, Minnesota (see figure 3.2). The energy spectrum of the

beam, shown in figure 3.3, allows experiments like NOνA to locate their detectors off the

1The number of protons per pulse is on the order of 1013 [72].2The power cited is the one delivered to the NOνA prototype detector. The MI is

currently in the middle of an upgrade to deliver 700 kW.

35

beam axis (see figure 3.4) to obtain narrower neutrino energy spectra.

Figure 3.4 Location Of The NOνA And MINOS Far Detectors. The MINOS fardetector in Minnesota is on-axis. Picture from Google Earth.

Protons from the MI are directed every 1.9 s, by single-turn extraction, into the NuMI

beamline. At 320 kW , there are 2.5 × 1013 protons delivered in spills of 8 µs, and focused

onto a target, producing secondary mesons, i.e. charged pions and kaons. For the neutrino

beam to be able to reach the Soudan MINOS far detector site, the proton beam is directed

downward at 58 mrad before it strikes the 0.95 m long NuMI graphite target, as shown in

figure 3.5. The forward-going mesons are focused and allowed to decay, producing the desired

neutrino beam. The focusing is performed by a set of two parabolic magnetic horns. The

charged pions and kaons selected by the horns propagate down a 675 m long (1 m radius)

decay tunnel. A hadron absorber (beam stop) is placed at the end of the decay tunnel

36

to remove the residual flux of protons and mesons, followed by a set of muon monitoring

detectors, as shown in figure 3.6 [71].

Figure 3.5 Schematics Of The NuMI Complex. Figure taken from [71].

Figure 3.6 NuMI Beam Schematics. Taken from [74].

In neutrino oscillation searches, the flavor composition of the neutrino beam should be

well known. The ideal case is to have a pure flavor beam, i.e. only one neutrino fla-

vor. The NuMI beam design goals include: achieving the highest possible νµ intensity,

low backgrounds from other neutrino flavors, well understood spectra to control system-

atic uncertainties, and the selection of neutrino energy spectrum matched to the oscillation

physics [75].

37

3.1 Primary Beam System

The process of producing the neutrino beam starts with a set of primary protons that hits

a fixed target, which produces interactions that yield the mesons that decay to produce the

neutrinos.

The primary protons are extracted from the MI ring and transferred through the ex-

traction enclosure through a steeply inclined carrier pipe to the target region located about

50 m underground. The extraction mechanism is a horizontal kick, leading to Lambertson

magnets3 deflecting the beam primarily in the vertical direction. The extracted protons are

focused and bent strongly downward by a string of quadrupoles and bending magnets so that

they enter the target hall located 122 m downstream of the extraction enclosure. Another

set of bending magnets brings the protons to the correct pitch, 58 mrad, for a zero targeting

angle beam directed toward the Soudan site [75].

To create the neutrino beam from the primary proton beam, these protons strike the

NuMI target to produce hadrons. This target is sufficiently long to enable most of the

primary protons to interact, as well as thin enough such that secondary interactions of the

charged pions and kaons are minimized by allowing them to escape through the sides, as

illustrated in figure 3.7. The depth of the field of the horn focusing system sets a limit on the

length of the target. The desired flux of charged pions and kaons out of the target decreases

with increasing radius due to particle re-absorption. The target stress due to the heat load

of the proton beam also decreases with increasing radius. As a consequence of the last two

statements, the target was designed to obtain the maximum yield and to ensure integrity

against mechanical failure [75].

3Lambertson magnets are used to separate two adjacent beams by providing a bendingfield for one beam and a field-free region for the other beam [76].

38

Figure 3.7 NuMI Target Drawing. Schematic drawing of a long and slim segmentedtarget designed for optimum production and decay of charged pions and kaons [75].

3.2 Secondary Beam System

The second stage in the production of the neutrino beam is to focus the mesons coming

out of the target in order to direct them toward the location of the detector. Therefore

the focusing horns produce toroidal magnetic fields and act as lenses to bend the secondary

particles back to the primary proton direction. The horns can either focus all the mesons at

a particular momenta or some of the mesons at all momenta. The parabolic shaped horns

produce magnetic fields that act as lenses, where the focal length is proportional to the

momenta of the mesons. The selection of a particular target position causes particles of a

certain momentum to be focused by the first horn. Mesons that were well focused by this

horn pass unaffected through a central aperture in the second horn. Other mesons move

to larger radii and are focused by the second horn, extending the momentum bite of the

system, as illustrated in figure 3.6. The horns are designed for 205 kA maximum current

39

pulses of 5.2 ms half-sine wave width, with a repetition rate of 1.87 s. The direction of the

current flowing in the horns determines which mesons are focused depending on the sign of

their electric charge. The forward horn current (FHC) focuses positive mesons which decay

into neutrinos, and the reverse horn current (RHC) focuses negative mesons which decay

into antineutrinos [75]. Figure 3.8 illustrates the direction of the two currents in the horns,

as well as the beam direction through the horns.

Figure 3.8 Beam And Electric Current Directions Through The NuMI Horns. Thedrawing illustrates the direction of the beam through the horn. The FHC (RHC) flows inthe direction of the blue (red) arrows inside the conductors that make the horns.

The decay tunnel design allows for a sufficient flux of νµ within the energy band required

by MINOS and NOνA. Alignment is paramount part of the tunnel’s design. Misalignment

of the decay tunnel along its length can not occlude the aperture of the tunnel by more than

2%. The volume of the decay tunnel is carefully chosen to reach a compromise between the

neutrino flux and the cost of construction4. The choice of tunnel’s radius balanced the loss

of secondary particles, that interact with the walls, with the cost of construction. A high

vacuum inside the tunnel is necessary to prevent unwanted interactions with air molecules.

The vacuum level in the tunnel ought to be 1 Torr or lower [75].

The main purpose of the hadron absorber is to eliminate hadrons that would overload the

4Pions of energies about 50 GeV have a mean decay length of several km.

40

data acquisition systems of neutrino experiments on-site. All primary protons that did not

interact with the target are absorbed at this stage. This absorber is not thick enough to stop

the muons present in the beam, which are undesired background to the measurements of the

neutrino experiments. These muons can be eliminated by providing sufficient material to

absorb their energy. The NuMI beamline is located in dolomite, which is a dense rock. The

340 m of dolomite between the end of the hadron absorber and the MINOS hall is sufficient

to stop all muons coming from the decay tunnel [75].

The neutrino beam monitoring systems enables the beam users to measure the quality

of the neutrino beam. The neutrino flux is monitored through measuring: the spatial distri-

bution of the hadrons directly upstream of the absorber, and the muons at various locations

within the dolomite shield, as seen in figure 3.6. In order to detect variations, the muon

intensity measurement is normalized to the number of incoming protons, and to each other,

while the measured profiles are compared to nominal profiles [75].

3.3 NuMI Beam Composition And Energy Spectrum

At 110 mrad

Neutrinos are produced in many weak processes, including particle decays. For NOνA,

the two most important sources of neutrinos are charged pion and kaon decays. The

charged kaons are mesons made of two quarks: K+: us, and K−: us, with mean life

time: τK = (1.2380 ± 0.0021) × 10−8 s, and mass: mK = (493.677 ± 0.016) MeV/c2

[77]. These have decay modes with only leptons (leptonic modes), with leptons and hadrons

(semileptonic modes), and with only hadrons (hadronic modes). The modes with the largest

41

branching ratios are5 [77]:

K± → µ± + νµ(νµ), leptonic mode, branching ratio: 63.55 ± 0.11%, (3.1)

→ π± + π0, hadronic mode, branching ratio: 20.66 ± 0.08%,

→ π0 + e± + νe(νe), semileptonic mode, branching ratio: 5.07 ± 0.04%,

→ π0 + µ± + νµ(νµ), semileptonic mode, branching ratio: 3.353 ± 0.034%.

About 96% of all the νµ, from charged kaon decays, reaching the NOνA prototype detector

at 110 mrad off the NuMI beam axis come from the leptonic mode. The rest of the νµ, at

the prototype’s location come from the semileptonic decay. A small νe component of the

FHC of the NuMI beam comes from the semileptonic charged kaon decays. At the NOνA far

detector, the knowledge of this νe initial component of the beam is extremely important since

it is the principal background to the νe appearance signal. There is no direct measurement

of the charged kaons flux from the NuMI beam, therefore a measurement that can constrain

the charged kaon production yield prediction could help to reduce the uncertainties in the

νe flux at 14 mrad.

The charged pions contain two valence quarks: π+: ud, and π−: ud, with mean life time:

τπ = (2.6033 ± 0.0005) × 10−8 s, and mass: mπ = (139.570 ± 0.00035) MeV/c2 [78]. The

decay mode with the largest branching ratio is6 [78]:

π± → µ± + νµ(νµ), leptonic mode, branching ratio: 99.98770 ± 0.00004%. (3.2)

The leptonic decay modes of charged kaons and pions are two body decays that produce

5νµ (νe) are associated with µ− (e−) and K−.6νµ are associated with µ− and π−.

42

(GeV/c)T

p0 1 2 3 4

Mes

ons

0

10

20

30

40

. NuMI MC.T

True Transverse Momentum, pπK &

K

π

(a)

(GeV/c)L

p0 5 10 15 20 25 30

Mes

ons

0

0.5

1

1.5

2

. NuMI MC.L

True Longitudinal Momentum, pπK &

K

π

(b)

Figure 3.9 Meson Transverse And Longitudinal Momenta. Predicted (a) transverseand (b) longitudinal momentum distributions of charged kaons (blue) and pions (red) thatdecay into neutrinos detected in the NOνA prototype detector. Momenta evaluated at thetime of production of the meson in the target. MC simulation.

muons and neutrinos isotropically in the center of mass reference frame. In this reference

frame the neutrino energies are fixed. The decaying meson is boosted to translate the decay

into the laboratory reference frame. As a result, the neutrino energies will be relatively broad

since these are now a function of the relativistic parameter γ = E/m. The neutrino energies

in this case are also a function of the angle θ between the momentum of the decaying meson

and the momentum of the neutrino. For small θ, i.e. highly relativistic mesons, the flux and

energy of the neutrinos are given by:

Φν =

(

2γ

1 + γ2θ2

)2 A

4πd2,

Eν =CmEm

1 + γ2θ2, (3.3)

where A is the transverse area of the detector that measures the neutrinos, d is the distance

between the decay point and the detector’s location, Cm is a constant that takes two values:

one for charged kaons, CK = 0.96, and one for charged pions, Cπ = 0.43; and Em is the

43

energy of the meson. Most of the mesons that yield the neutrinos seen at the location of the

prototype detector have transverse momenta (pT ) of the order of 300 MeV, as seen in figure

3.9a. The longitudinal momenta (pL) of the mesons is mostly below 10 GeV/c, as seen in

figure 3.9b.

Energy (GeV)ν0 1 2 3 4 5

Eve

nts

/ 0.1

GeV

310

0

5

10

15

20

True Energy. Mother Particle. NDOS MC.ν

All±K

±πµ

LK

Figure 3.10 Neutrino Energy Distribution Discriminated By The Mother ParticleOf The Neutrino. All mother particles (black), π± (red), K± (blue), KL (light blue), andµ± (green) energy distributions. MC simulation.

The neutrino energy spectrum observed with the prototype detector shows two peaks: one

comes from charged pion decays, around 0.2 GeV, and the second resulting from charged

kaon decays, around 2 GeV. The spectrum is produced mostly by νµ. However this also

receives contributions from νµ, νe, and νe which arise from the semileptonic charged kaon

decays, the leptonic decay of K− that were not defocused by the horns, and muons and KL

decays. 60.2% of all neutrinos come from charged kaon decays, 36.6% come from charged

pion decays, 2.9% resulting from KL decays, and 0.2% arising from decayed muons, as shown

in figure 3.10. The 4 neutrino components of the predicted energy spectrum, shown in figure

44


Eve

nts

/ 0.1

GeV

310

0

5

10

15

20

Types. NDOS MC.ν True Energy. All ν

Allµν

µνeν + eν

Figure 3.11 Neutrino Energy Distribution Discriminated By The Neutrino Type.All neutrino types (black), νµ (blue), νµ (red), and νe + νe (green) energy distributions.MC simulation.

3.11, are distributed as follows: νµ 81%, νµ 14.3%, νe 3.9%, and νe 0.8%. The ratio Kπ as

a function of energy is shown in figure 3.12. For energies higher than 1.8 GeV the ratio

remains above 10. This motivates a measurement of the neutrino flux coming from charged

kaon decays due to the low background from neutrinos produced in other decays.

45


Rat

ioπ

K/

-210

-110

1

10

210

Ratio. NDOS MC.π True Energy. K/ν

Figure 3.12 Ratio Of Neutrinos From Charged Kaon Decays Over Neutrinos FromCharged Pion Decays. MC simulation.

46

Chapter 4

The NOνA Prototype Detector

4.1 Experiment Overview

The NOνA experiment will use the NuMI beam to measure νµ → νe oscillations. The

neutrino energies at the NuMI beam are a function of the energy of the meson, as well as of

the angle between the momentum of the meson that produces the neutrino and momentum

of the neutrino, as seen in equation (3.3). The MINOS experiment, located on the NuMI

beam axis, measured neutrinos on a wide energy range [79] (see figure 3.3). The NOνA

experiment, 14 mrad off the NuMI beam axis, will measure neutrinos at a much narrowed

energy band centered at 2 GeV. The far detector’s location, 810 km away from the neutrino

source (see figure 3.4), and the narrow neutrino energies observed there, produce the ideal

conditions to observe the νµ → νe oscillation very close to the maximum of probability.

Using equation (2.49), this maximum occurs at:

2chπ

∆m232

=L

E≈ 534

km

GeV, (4.1)

while NOνA has L/E = 505 km/GeV. Under such circumstances, NOνA will be capable of

determining the mass hierarchy and the CP violation with a significance up to 2σ, depending

on the real value of the oscillation parameters1.

1A sample bi-probability point fulfilling the previous statement is shown in figure 4.1.

47

Figure 4.1 Bi-probability Plot. Four values of the phase δ are presented of each of the masshierarchies: normal (blue) and inverted (red). Official NOνA figure [81]. 1σ and 2σ contoursare drawn around the estimated bi-probability points. The lower right point representedby a star is at δ = 3π/2. The upper left point represented by a star is at δ = π/2. MCsimulation.

48

NOνA plans to collect three years of data in each mode2. From each mode, the transition

probabilities (recall equation (2.53)): P (νe) for νµ → νe, and P (νe) for νµ → νe, will

be estimated. Once these two probabilities are calculated, a point can be drawn in a bi-

probability plot [80] like the one shown in figure 4.1, and the mass hierarchy and the phase δ

could be resolved. 1σ and 2σ contours are shown around the calculated points. The ellipses

are representations of the various possible values of δ. The maximum CP violation occurs

at δ = π/2, 3π/2, while no CP violation occurs for δ = 0, π. Each mass hierarchy has

its own ellipse due to matter effects. The bi-probability plots summarize all the oscillation

parameters in an instructive fashion. Figure 4.1 assumes θ23 = π/4. Should θ23 > π/4, the

ellipses will move towards higher P (νe) values along the P (νe) = P (νe) line. On the other

hand, should θ23 < π/4, the ellipses will move downwards to lower P (νe) values along the

P (νe) = P (νe) line. The best case scenario for NOνA would be a normal hierarchy and

δ = 3π/2, or an inverted hierarchy and δ = π/2. These two points will exclude the other

mass hierarchy with a 2σ significance.

The measured value of θ13 (see table 2.1) gives NOνA the best sensitivity to: resolve

the mass hierarchy, measure δ, and resolve the octant of θ23. In order to make these mea-

surements, NOνA built two detectors. The near detector at Fermilab has a 3.9 × 3.9 m2

transverse area, and is 14.3 m long. The far detector has a 15.6 × 15.6 m2 transverse area,

and is 63.0 m long. Figure 4.2 shows to scale the three detectors. An Airbus A380-800 is

included in the figure for scale. The far detector is the largest free-standing PVC structure

in the world.

2Neutrino and anti-neutrino modes.

49

Figure 4.2 Drawing Of The NOνA Detectors. NOνA detectors drawn to scale. Anaverage height person is shown to scale. An Airbus A380-800, drawn to scale, is includedfor illustration purposes [82].

50

4.2 The Near Detector On The Surface

(mrad)NuMIθ 0 200 400 600 800

Eve

nts

3 1

0

-310

-210

-110

1

10

210

Angle From NuMI Beam. NuMI MC.ν

(a)

(mrad)NuMIθ 0 200 400 600 800

Eve

nts

3 1

0

-310

-210

-110

1

10

.±π & ± Parent: Kν Angle From NuMI Beam. NuMI MC. ν

±π Normalized to ±K

±K±π

(b)

Figure 4.3 Angle Of Neutrinos With Respect To The NuMI Beam. (a) θNuMI forneutrinos from all parents. (b) θNuMI for neutrinos from charged kaons (blue) and chargedpions (red) parents. The charged kaons area is normalized to that of the charged pions. MCsimulation.

The NOνA prototype detector is located on the surface, north from the MI at Fermilab

(see figure 3.1), and 110 mrad off the NuMI beam axis (see figure 3.5). This angle is measured

from the NuMI target to the origin of coordinates of the prototype detector. Not all neutrinos

are created at the same location along the decay pipe, their angular distribution is shown in

figure 4.3a for all neutrino parents. About 77% of all these neutrinos have:

110 mrad < θNuMI < 120 mrad, (4.2)

The angular distributions for neutrinos from charged kaon and pion decays are shown in

figure 4.3b. Neutrinos from charged pion decays have a slightly higher rate between:

120 mrad < θNuMI < 200 mrad, (4.3)

51

when compared with neutrinos from charged kaon decays. The peak of neutrinos around

650 mrad, in figure 4.3, comes predominantly from low longitudinal momentum mesons

(about 75% of which are charged kaons decaying into a νµ and a muon), as seen in figure

4.4a. The transverse momentum of these mesons is shown in figure 4.4b. Low energy

neutrinos make this peak, as seen in figure 4.4c.

(mrad)NuMIθ 0 200 400 600 800

(G

eV/c

) L

P

0

5

10

15

20

25

30

1

10

210

310

Angle From NuMI Beam. NuMI MC.ν vs. LMeson’s P

(a)

(mrad)NuMIθ 0 200 400 600 800

(G

eV/c

) T

P

0

1

2

3

4

1

10

210

310

410

Angle From NuMI Beam. NuMI MC.ν vs. TMeson’s P

(b)

(mrad)NuMIθ 0 200 400 600 800

Ene

rgy

(GeV

) ν

0

2

4

6

8

10

1

10

210

310

: Energy vs. Angle From NuMI Beam. NuMI MC.ν

(c)

Figure 4.4 Meson Momentum And Neutrino Energy As A Function Of The NuMIAngle. (a) Charged pions and kaons longitudinal momentum as a function of θNuMI, (b)charged pions and kaons transverse momentum as a function of θNuMI, and (c) neutrinoenergy as a function of θNuMI. MC simulation.

The initial purpose of this detector was to serve as a prototype, and then be moved

underground to become the near detector. For this reason it is known as the Near Detector

On the Surface (NDOS). However, the collaboration decided to build a totally new and

52

improved near detector. As a result the NDOS now serves as a testing facility for various

hardware and software purposes.

Figure 4.5 Drawing Of The Prototype Detector. The Prototype detector is a collectionof six blocks, each made of 31 planes. The neutrino beam goes, in the direction indicatedby the black arrow, upstream to downstream. At the end of the six blocks there is amuon catcher that helps to reconstruct the energy of the most energetic muons. Artisticrepresentation.

The NDOS is 3.9 m high (Y ), 2.6 m wide (X), and 14.3 m long (Z), as shown in figure 4.5.

It is a modular structure made from highly reflective PVC extrusions [83]. Each extrusion

has 16 cells, and each cell has a 3.9 × 6.0 cm2 transverse area (see figure 4.6 for an illustration

of a single cell). Inside each cell there is a looped wavelength shifting (WLS) fiber, both

ends of the fiber are placed against the face of an avalanche photodiode (APD), and the

fiber covers the full cell length laid in a U-shape. Extrusions are arranged in modules, there

are two extrusions per module. Modules are arranged in planes, there are two (X) or three

(Y ) modules per plane. A block is an array of interspersed vertical (16) and horizontal (15)

planes. This arrangement of interspersed planes gives the X − Y coordinate system of the

53

Figure 4.6 NOνA Extrusions And Cells. Sample NOνA PVC extrusions arranged in amodular collection of vertical and horizontal planes. Artistic representation of a single cell[82].

Figure 4.7 Side View Of The Prototype Detector. Horizontal planes readouts.Avalanche photodiode, front-end board and data concentrator modules [84].

54

detector. The NDOS bulk is 6 blocks long. Downstream of the detector there is a muon

catcher, which is a collection of interspersed extrusions and ten 10 cm thick steel plates. At

the end of the muon catcher, four more planes are positioned in the usual vertical-horizontal

arrangement used for containment purposes, as seen in figure 4.5.

Each APD contains 32 channels or pixels, where each pixel is exposed to both ends of a

single fiber in a cell. Each APD is mounted on a front-end board (FEB) which reads and

digitizes the signal3. The digital output of each FEB is sent to a data concentrator module

(DCM) to consolidate the information. Each DCM reads 64 FEB (figure 4.7 illustrates the

arrangement of the various hardware components). All the electronics described above are

mounted on the west side of the detector, as seen in figure 4.7, for the horizontal planes, and

on top of the detector for the vertical planes.

The segmented nature of the NOνA detectors allows for multiple sampling of the energy

depositions of muons and electromagnetic showers4. The radiation length of the detectors

(X0) is X0 ≈ 40 cm, and the Moliere radius is RM ≈ 10 cm. The X0 of the detectors allows

to get a very detailed sample of 2 GeV electron showers which are, on average, about 9X0

in length [85]. Photon showers are, on average, about 9X0 long as well, however the gap

these leave between the vertex and the start of the shower, which is X0 long, provides an

element of discrimination. With about 7 cells per X0, the energy depositions are extremely

useful in particle identification. This same powerful feature is used in muon identification.

The signature of a νµ event in the detectors is a muon, and the experimental setup of NOνA

allows to identify and study them with great precision and detail. Identifying electrons is

extremely important for the νe appearance measurement. A background to electrons are the

3See section 4.2.4.4The segmented geometry of the NOνA detectors is optimized for the identification and

measurement of νe CC interactions, as discussed at the end of section 4.2.

55

photons that result from π0 decays.

4.2.1 Liquid Scintillator

The primary target in the detectors is the carbon nucleus. About 80% of the carbon nuclei

in the detectors come from the liquid scintillator (LS) blend that fills the PVC extrusions.

This blend is 95.79% mineral oil, used as a solvent, 4.11% scintillating pseudocumene, and

0.10% waveshifters [86]. When ionizing particles traverse the LS, ultraviolet light is emitted

[87]; the emitted wavelengths are then shifted by the waveshifters to the violet-blue range.

This light bounces, on average, 10 times inside a cell before it is captured by the WLS fiber.

In order to avoid light losses between emission and absorption, the PVC walls have 12%

TiO2 which gives a reflectance of about 90% for green wavelengths [21].

4.2.2 Wavelength Shifting Fiber

Wavelength shifting fibers are a critical component in the NOνA detectors. These collect

light emitted from the LS and transmit it into the APDs. Inside the far detector’s cells

the light travels as much as 16 m, which requires shifting of the wavelengths from the short

attenuation lengths of the blue (about 4 m) to the longer attenuation lengths of the green

(about 15 m). In addition, the APDs are more efficient in the green than in the blue light.

The spectra of the light emitted by the fiber as a function of the illumination point is

presented in figure 4.8. The highest light power is the spectrum for the light shined into the

fiber at 1 m from the readout. The lowest light power is the spectrum for the light shined

into the fiber at 26 m from the readout. Intermediate spectra are for illumination points at

1 m intervals. The various spectra show how the peak of the distribution rapidly transforms

56

Figure 4.8 Emission Spectra Of The Wavelength Shifting Fiber. Emission spectrafor the WLS fiber at various illumination points [88].

57

Figure 4.9 Wavelength Shifting Fiber Transverse Area.

from the 470 nm of the blue wavelength to the 580 nm of the green as the illumination point

moves father apart from the readout.

The WLS fiber used by NOνA is 0.7 mm in diameter (D). This features a polystyrene

(PS) core with a refractive index of 1.59, surrounded by a polymethylmethacrylate (PMMA)

inner cladding with a refractive index of 1.49, and a fluorinared polymer (FP) outer cladding

with refractive index of 1.42, as shown in figure 4.9. This multicladding structure produces

50% higher yield than a single cladding fiber due to a large trapping efficiency [89].

4.2.3 Avalanche Photodiodes

The NOνA collaboration selected APDs as photodetectors due to their high quantum effi-

ciency and low cost. The 85% quantum efficiency enables the use of the long cells in the

far detector5. The average signal for muons traversing a cell perpendicular to its walls is

5Photomultiplier Tubes have a 10% quantum efficiency under similar circumstances.

58

30 photons at the face of the APD, which gives about 25 photoelectrons. This signal must

be distinguishable from high frequency noise. APDs are made from silicon, which features

a thermal generation of electron-hole pairs that mimic the signal from the fibers. These

electrons are amplified at the diode junction and appear as the input to the pre-amplifier,

thus contributing directly to the noise. In order to achieve a signal to noise ratio of 10:1,

the APDs are cooled to -15 oC.

Figure 4.10 Avalanche Photodiode.

The light output of the fiber is absorbed in the APD collection region, shown in figure

4.10, where electron-hole pairs are generated and, under the influence of an applied electric

field, the electrons propagate to the p-n junction. At the junction, the electric field, which

determines the multiplication of the current, is high enough to produce avalanche multipli-

cation of electrons. The current generated from these electrons is the output signal of the

APD [21].

59

4.2.4 Front-End Boards

The readout of the APD, at the FEB, requires a pre-amplifier that can sample the signal

throughout a 10 µs time window. The signal from the APD is amplified and shaped by a

high-gain integrating amplifier with a shaping time of about 350 ns, and the output is stored

in a switched capacitor array (SCA) every 500 ns. The SCA contains 64 samples taken

500 ns apart for the 32 channels in the APD. The waveform is designed to have a 380 ns rise

time and 7000 ns fall time [90]:

F (t) ∝(

1 − e−t/380ns)

e−t/7000ns. (4.4)

These choices of rise and fall times are intended to minimize the overall noise induced by the

electronics. Zero suppression is performed at 15 - 20 photoelectrons6 via a dual correlated

sampling (DCS) algorithm which subtracts the signal from the baseline. The DCS establishes

a rising edge triggered threshold under which the sampling points are zero suppressed. The

DCS algorithm then subtracts the sample taken right before the threshold is crossed from the

sample, with maximum amplitude, taken right after the threshold is crossed. This threshold

is set independently for each channel of the detector. Fits to the shape of the waveform

recover the timing and pulse height information [91].

The best timing resolution with this system is 500/√

12 ns ≈ 145 ns. The digitized

differences are stored in a field programmable gate array (FPGA) for transmission to the

data acquisition (DAQ) system. Digital signal processing algorithms could be encoded in the

FPGA firmware to improve both the signal-to-noise ratio and timing resolution. A FPGA

6ADC and photoelectrons are proportional. See section 4.2.6 for a discussion on energycalibration.

60

produces timing markers at periodic intervals of 50 µs interspersed with digitized hits [92].

4.2.5 Data Acquisition System

A DCM is a custom electronic component used to consolidate and concentrate the data

from 64 FEBs. A DCM programs, configures, and monitors each FEB, and passes the

timing system clock (TSC) and synchronization command to each FEB. The DCM-FEB

communication is through point-to-point serial data links with dedicated differential pair

lines for clock, synchronization, command and data. The digitized hits are consolidated by

the DCM FPGA to 50 µs time slices containing data from all 64 FEBs. An application

further consolidates this data to a longer 5 ms time slice and routes this time slice to a

downstream buffer node for processing. All DCMs throughout the detector send hit data

corresponding to one 5 ms time slice to a buffer node [92].

A buffer node is a commodity server residing in a 140 buffer nodes buffer farm. An

external global trigger system provides triggers to a buffer node in the form of a start

time and a time window, so that an application can select data corresponding to this time

window from buffered data. Only trigger-selected data are written by the buffer nodes to a

downstream data logger.

A data logger is a program that receives trigger-selected data from the buffer farm to

format into an event. A trigger issued to the buffer farm is also issued to the data logger

application for validation purposes. The event created by the data logger is written to disk,

and then archived to Fermilab’s mass storage. Events are also written by the data logger

to a shared memory segment, from which a dispatcher application can serve the data to a

quasi-online external client application.

All FEBs and DCMs throughout the DAQ system are synchronized through the use of

61

Figure 4.11 Schematic Overview Of The Data Acquisition System [92].

62

a sophisticated timing system. This system provides a stable master clock line that permit

the time stamp counters that are present on the FEBs, DCMs, and the timing system to

be loaded and synchronized with a universal wall time based off of a link to the GPS. The

timing system can achieve unit to unit synchronization that is accurate to within one clock

cycle. This system is also used to time stamp the beam spill information coming from the

Fermilab accelerator complex [91].

Each timing distribution unit (TDU) has four differential pair lines for communication:

synchronization, command, clock, and synchronization echo. The synchronization echo line

is used to compensate for cable length propagation delays. Refer to figure 4.11 for a DAQ

system schematic that illustrates the various steps and components involved in the data

processing.

Individual data output files are saved by SubRuns, which are either an hour long, or

512 MB in size, which ever happens first. SubRuns are grouped in Runs which can have up

to 24 SubRuns. Any time that the DAQ systems are stopped, that is the end of the SubRun

and Run in progress. The SubRuns that recorded the NuMI spills have one event per spill,

and each event is 500 µs long [92].

4.2.6 Performance And Calibration Of The Prototype Detector

The NDOS started its physics run in October 2010 during which cosmic ray data, and νµ and

νµ data from the NuMI beam were collected. The collaboration intended to fully instrument

the detector with APDs. However, various problems occurred during commissioning, and the

majority of the APDs were damaged due to moisture on their face coming from condensed

water deposited when the detector was cooled down. About a third of the APD set survived

the problems. The APDs that survived were strategically located throughout the detector in

63

Figure 4.12 Event Display Of The Prototype Detector. Vertical planes make the topview, and provide the horizontal (X) coordinate. Horizontal planes make the side view, andprovide the vertical (Y ) coordinate.

order to optimize the observation and measurement of neutrino interactions in the detector.

Most of the available APDs were gathered together in a region in order to identify the

neutrino interaction point, and the rest of the available APDs were located in strategic

individual planes in order to record the full length of the muon tracks. The plan with the

muon catcher was to have it fully instrumented. However there are some channels maps with

some regions that were also partially instrumented.

An event display of the detector is presented in figure 4.12. Cells with active APDs are

drawn with black lines, while inactive cells are drawn in light grey. The top view is made

from vertical planes which provide the X coordinate; the side view is made from horizontal

planes which provide the Y coordinate. Neutrinos from the NuMI beam enter the detector

from the bottom (Y = −200 cm). The bulk region, where most of the APDs are located, is

centered at Z = 400 cm. If APDs became available, these were installed to improve muon

identification, and if APDs failed, these were removed. This configuration is known as NDOS

run II, and the data recorded under this configuration goes from October 2011 until April

64

2012, during which there were 1.67 × 1020 protons on target (POT). The APDs operated

warm to avoid further losses. The active channels map for each SubRun is known, and in

general, such maps change frequently.

Figure 4.13 Cosmic Data Of The Prototype Detector. Distance from the center ofthe detector vs. mean ADC value per cm. Attenuation corrections come from the fit. Thereadouts are in the right. Official NOνA NDOS figure [81].

The performance and calibration of the detector were initially studied using cosmic ray

data. The energy calibration procedure used energy deposition of cosmic rays at different

locations in the detector. The light level is measured, initially, in analog to digital converter

(ADC) units of collected charge. Apart from a drop in the average light at short distances

from the APD (near the beginning of the PVC cell), the mean observed light level decreases

with increasing distance of the hit from the APD, as seen in figure 4.13. Here, the position

(W ) of a hit is measured from the center of the detector. The electronics are beyond the

W = 200 cm point. The goal of the energy calibration is to make hits depositing the same

energy, but located at different distances from the readouts, result in the same corrected

65

energy measurement. The mean ADC/cm is a representation of the light attenuation per

cell. The rapid fall at Y < −140 cm is due to the sum of two exponential functions that

represent the light attenuation in the fiber [93]:

A(r) ∼ ASe−r/RS + ALe

−r/RL . (4.5)

Here r is the distance traveled by the light inside the fiber, RS = 289.5 cm is a short

attenuation length, and RL = 852.3 cm is a long attenuation length. Light entering the fiber

is split up according to the constants AS = 0.3137 and AL = 0.1669, and propagated to the

readouts according to the two attenuation lengths [93]. The rapid fall at Y > 140 cm is due

to light absorption at the manifold that guides the fibers to the APDs.

(a) (b)

Figure 4.14 Energy Calibration Of The Prototype Detector. ADC distributions forvarious W slices (a) before and (b) after attenuation corrections. Official NOνA NDOSfigure [81].

Muons passing through the detector exhibit an uncorrected light level distribution with

a mean that shifts from about 400 ADC, at the closest distance to the readouts, to about

300 ADC at the farthest distance to the readouts, as shown in figure 4.14a. The small peak

at low light levels is the tail of the APDs noise distribution above threshold, and is due to

66

cells in which the muon passes entirely through the detector PVC. After the attenuation

corrections are applied the light level distributions overlap, as seen in figure 4.14b, so that

the energy measurement is then independent of W .

To approximate to photoelectron (PE) units, a simple rescaling of ADC is carried out:

ADC/PE = 2.275 [93]. The linear relation between ADC and PE is seen in figures 4.15a

and 4.15b, where the same conversion factor appears for the MC simulation and the data,

respectively. Application of the position dependent corrections discussed above yielded a

corrected number of photoelectrons (PECorr). The corrections are a function of W , which

means that ADC and PECorr are not proportional, as seen in figures 4.15c and 4.15d. The

initial assumption is that at W = 0: PE = 1 = PECorr [93]. The poor χ2 tests indicate

that the two quantities do not follow a straight line correlation.

To convert from PECorr to an absolute energy deposition, the range-energy relationship

provided by the Bethe Bloch formula [87] is used. A sample of cosmic ray tracks that clearly

stopped within the detector is selected and the reconstructed stopping point determined

for each track. With the distance (dc) from the center of each cell in the track to its

endpoint, a relation between PECorr/cm and dc is established. The Bethe Bloch curves are

integrated to provide the expected dE/dX for every dc. The integration takes into account

the density of the material that receives the deposited energy, i.e. to distinguish between

PVC, scintillator, etc.. The best fit to the profile of the PECorr/cm plot yields a calibration

of: GeV/PECorr = 4.7807 × 10−5 [93], about 50 keV per corrected photoelectron.

The performance of NDOS over time is represented by the light level in a set of test cells.

The set of cells is located at the center of the detector in X -Y and at the south edge of the

bulk of APDs, Z = 280 cm. During the running of the NDOS there were times with the

APDs system cold or warm, and FEBs gains were varied; however, the performance of the

67

(a) (b)

(c) (d)

Figure 4.15 Energy Conversion Factors Of The Prototype Detector. ADC vs. PEfor (a) MC simulation and (b) data. ADC vs. PECorr for (c) MC simulation and (d) data[94].

68

Figure 4.16 Light Level Over Time On The Prototype Detector. Light level on 8 testchannels of an APD as a function of time [95].

detector remained stable, as seen in figure 4.16. The time elapsed starts six months prior to

the NDOS run II, and ends a month before the end of this run. During the period shown,

there were times of cold and warm APD, and the gains varied in the FEB; however, the

performance of the detector remained stable.

4.3 NOνA Software

To evaluate the biases affecting various physics process involved in the analyses, the NOνA

collaboration developed a Monte Carlo (MC) detector simulation, and track reconstruction

programs which could be applied to both MC and data objects.

69

4.3.1 Monte Carlo Simulation

The NOνA simulation process begins at the NuMI beam level. The MC simulates the hadron

production in the NuMI target, which is followed by the propagation of the produced particles

through the target material, the magnetic horns and the decay pipe. The FLUKA software

[96] simulates interactions of 120 GeV protons in the NuMI target, and the interaction

of secondary particles created in the target as well. The simulation creates output files,

which contain the kinematic variables of the generated particles. These files are input to

the GEANT4 [97] software, which propagates the particles, and their decay products, from

the NuMI target, through the magnetic horns, and down the decay pipe, where weak decays

yield neutrinos. If the combination of FLUKA and GEANT4 (known as FLUGG [98])

generates a neutrino that reaches the NDOS, the GENIE [99] software simulates interactions

in proportion to predetermined cross sections. The interaction is simulated within the target

nucleus, and an intranuclear hadron transport yields identified final state particles with full

kinematics.

The NOνA collaboration has developed a full simulation of the NDOS using GEANT4 to

propagate GENIE generated particles through the detector. The software simulates energy

deposition, scattering, and decay processes that affect the particles produced during neutrino

interactions. The NOνA collaboration developed further simulation tools that use the energy

depositions in the scintillator to generate blue light, reflect it in the PVC cells until absorbed

by wavelength shifting fibers. Equation (4.5) parametrizes the attenuation of the green

shifted light as it propagates through the fibers to the APD and the readouts. All the values

of variables created by the MC simulations that are available for analysis are named: true

values.

70

To provide samples with the same biases as the real data, the MC SubRun are created

with the same active channels maps. Also, the number of POT used to produce the MC

sample is stored in the files, as it is done in the data.

4.3.2 Reconstruction Tools

The reconstruction tools developed by the NOνA collaboration are intended to work with

simulated and real values of the variables7. The basic unit in the reconstruction process is a

hit. A hit is an energy deposition left in the detector by a particle. The simulation provides

hits for every interaction of a particle within the detector; there could be many simulated

hits inside an individual cell. Data also provides hits, which are energy depositions in a

detector cell that generate a detected amount of light, above the electronic noise threshold,

at a digitized time, and with a position provisionally given as the center of the cell. Any

group of real hits within a cell that share the same time tag will be collected into one hit:

a cell hit. Cell hits appear both in simulated and real data of the variables. Cell hits have

stored values of: energy, time, and position; which have been properly calibrated (see section

4.2.6).

4.3.2.1 The Slicer

In order to reconstruct a neutrino interaction, whether it is simulated or data, the first stage

in the reconstruction is to make slices of cell hits that share common features in space and

time. The Slicer sorts the cell hits in an event by time, after which subsets of cell hits (slices)

that are grouped by similar times are produced. Different slices are separated by at least

7Real values of the variables are those provided by the DAQ, and come from real mea-surements, or data.

71

Figure 4.17 Sample Cosmic Ray Event For Slicer Performance. Colors represent thevarious physics slices.

530 ns. A physics slice is one where the number of cell hits is eight or greater.

Cell hits are considered noise, and removed from a physics slice, if they are isolated from

other cell hits in space and/or time. Cell hits with PE < 15 are also considered noise, and

removed from the physics slice: noise subtraction. Cell hits subtracted from a physics slice

are not recorded as noise, but are incorporated to the noise slice. After the classification

done by the Slicer there would be: physics slices, and a noise slice that collects all the noise

hits in the event. In events defined by a 500 µs time window, the performance of the Slicer

is determined by its efficiency (ǫ) and purity (p):

ǫ =stse

= 96%, p =stsp

= 99%, (4.6)

where se is the total number of true cell hits in the event that are not labeled as noise, st is

the total number of cell hits in the physics slices and not labeled as noise, and sp is the total

number of cell hits in the physics slices [100]. Colored dots are drawn on top of the cell hits

72

to represent each of the various physics slices, as shown in figure 4.17, for a sample cosmic

ray event. The noise slice is not drawn.

4.3.2.2 Track Reconstruction

In νµ CC interactions, the nominal path of the muon through the detector is a straight line,

though multiple scattering can cause noticeable deviations, especially near a stopping point.

Nevertheless, tracking muons in the detector is straightforward, and their initial energies

can be calculated from the paths lengths, if the muons stop in the detector. The NOνA

collaboration developed a series of tracking algorithms to reconstruct cosmic tracks, tracks

from neutrino interactions, and showers. In tests on MC simulated muon tracks in the NDOS,

with its large number of inactive channels, the Kalman Tracker algorithm performs better

than the alternatives at reconstructing muon tracks produced by neutrino interactions.

To test the performance of the tracker on various kinds of particles, the difference between

the true and the reconstructed track lengths (∆L) divided by the true track length (L) is

examined. The test shows that the tracker is most efficient at reconstructing muon track

lengths, and not so efficient at reconstructing the lengths of other particles. This test is most

relevant to this analysis of νµ CC interactions, where tracking is used to identify a candidate

muon track. The track length allows the energy of the muon to be calculated. The remainder

of the neutrino interaction energy is determined using the detector as a calorimeter. The

overall efficiency (ǫ) and purity (p) of the tracker are:

ǫ =kt

ke, p =

kt

kp, (4.7)

where ke is the total number of true cell hits that belong to a particle, kt is the total number

73

of cell hits in the reconstructed track that belong to that particle, and kp is the total number

of cell hits in the reconstructed track. The efficiency and purity for muons are higher than

for protons and charged pions. A summary of the performance of the tracker is presented in

table 4.1. For a complete study on the performance of the various NOνA trackers see [101].

Kalman Tracker All µ p π±

Efficiency 75% 92% 69% 68%Purity 61% 87% 37% 41%

100% × ∆L/L - 23% 131% 78%

Table 4.1 Kalman Tracker Performance. Kalman Tracker’s efficiency, purity and ∆L/Lfor various particles [101].

The Kalman Tracker uses a Kalman filter [102] for track pattern recognition and track

fitting. In this application, the filter has the ability to obtain pattern recognition and track

fitting in one module, and the capability to find multiple tracks within a group of correlated

cell hits. The input of the filter are slices from the Slicer, and the output are 2D and 3D

tracks. The 2D track reconstruction is performed on each physics slice, and separately in

each view. Whenever possible, the 2D tracks found in separate views are matched together

to form 3D tracks if these belong to the same physics slice. The Kalman Tracker algorithm

is described in appendix B.

The Kalman Tracker does not reconstruct a vertex. Therefore the interaction vertex is

placed at the starting (lowest Z coordinate) position of the longest track in the neutrino

interaction. The vertex resolution is very good in the three coordinates, about a cell, as seen

in figure 4.18. The tails in the distributions (beyond 1σ) account for 15% of all the tracks.

These tails are due to the large number of inactive channels in the detector, but they are

sufficiently small to make a reasonable fiducial volume cut based on the coordinates of the

vertex.

74

Mean 0.2717

RMS 11.76

Constant 7.882e+04

Mean 0.05367

Sigma 3.344

x (cm)∆ -100 -50 0 50 100

Tra

cks

0

20

40

60

80310×

Mean 0.2717

RMS 11.76

Constant 7.882e+04

Mean 0.05367

Sigma 3.344

x. NDOS MC.∆Vertex Resolution in x:

(a)

Mean -0.01085

RMS 11.87

Constant 7.652e+04

Mean 0.01203

Sigma 3.422

y (cm)∆ -100 -50 0 50 100

Tra

cks

0

20

40

60

80310×

Mean -0.01085

RMS 11.87

Constant 7.652e+04

Mean 0.01203

Sigma 3.422

y. NDOS MC.∆Vertex Resolution in y:

(b)

Mean 1.432

RMS 13.63

Constant 5.803e+04

Mean -0.178

Sigma 4.264

z (cm)∆ -100 -50 0 50 100

Tra

cks

0

10

20

30

40

50

60310×

Mean 1.432

RMS 13.63

Constant 5.803e+04

Mean -0.178

Sigma 4.264

z. NDOS MC.∆Vertex Resolution in z:

(c)

Figure 4.18 Vertex Resolution. (a) X, (b) Y , and (c) Z. MC simulation.

75

Entries 159835Mean -0.4485RMS 48.22Underflow 3.347e+04Overflow 3.276e+04Integral 9.36e+04Constant 1.308e+04Mean -0.008798Sigma 8.097

x (cm)∆ -200 -100 0 100 200

Tra

cks

0

5

10

15

310×


x. NDOS MC.∆End Point Resolution in x:

(a)


y (cm)∆ -200 -100 0 100 200

Tra

cks

0

5

10

15

310×Entries 159835Mean -15.86RMS 56.29Underflow 1.595e+04Overflow 4.674e+04Integral 9.714e+04Constant 1.197e+04Mean -0.4022Sigma 7.899

y. NDOS MC.∆End Point Resolution in y:

(b)

Entries 159835Mean 11.4RMS 68.94Underflow 8656Overflow 6.61e+04Integral 8.508e+04Constant 9669Mean 2.207Sigma 7.133

z (cm)∆ -200 -100 0 100 200

Tra

cks

0

2

4

6

8

10

12

14310×

Entries 159835Mean 11.4RMS 68.94Underflow 8656Overflow 6.61e+04Integral 8.508e+04Constant 9669Mean 2.207Sigma 7.133

z. NDOS MC.∆End Point Resolution in z:

(c)

Figure 4.19 Endpoint Resolution. (a) X, (b) Y , and (c) Z. MC simulation.

76

Finding the endpoint of the longest track has some significant problems. The large gaps

in Z, with no active channels in the detector downstream of the interaction region, presents

a serious obstacle to achieving acceptable performance with the tracker. Therefore, the

resolution for the endpoint of the longest track is not as good as the vertex resolution. When

muons reach the regions where only a few planes have active channels (Z > 600 cm), the

endpoint of long tracks has a large uncertainty, as seen in figure 4.19. The long tails in figures

4.19a, 4.19b, and 4.19c account for: 63.2%, 66.4%, and 75.7% of all tracks, respectively. The

overall endpoint resolution in X and Y is about 50 cm, and in Z is about 60 cm. For

reference, a 2 GeV muon deposits 120 MeV of energy in 60 cm of scintillator.

Figure 4.20 Sample Event. Case (1). MC simulation.

Reconstructed tracks have their starting and ending points in the center of an active cell.

However, a real muon is not guaranteed to end its trajectory in an active cell due to the

distribution of active cells. The trajectory of a long muon is more likely to end in an inactive

cell. The tracker places the end of the track in the last active cell with energy deposition

77

along the track’s path. In the following cases, however, the real endpoint of the muon would

not be accurately reconstructed:

1. The muon could have ended its path in the region between planes with active cells.

2. The muon could have passed a plane with active cells through the PVC.

3. The tracker could have missed a cell hit in an active plane.

4. The muon could have left the detector before reaching the next active plane.

About 30% of the muons exhibit one of these behaviors [103]. In case (1), shown in figure

4.20, the muon stops at the end of the dotted path in red. On average, however, a muon

with this topology will have an endpoint halfway between the end of the dotted path and the

blue “x”, as illustrated by the magenta line segment added to the end of the reconstructed

track (red). The length of this segment extends the track to a point half the distance (H)

between the adjacent active planes, so that H/2 is added to the Z coordinate of the track’s

endpoint. With this correction, the difference between the true and reconstructed endpoint

is, on average, centered at zero, and exhibits a resolution of 25 cm, or the thickness of 5 cells

[103].

The other three cases are more difficult to accurately reconstruct. Sample events of the

last three cases are presented in figure 4.21, where top views for each event are labeled: Reco

for the reconstructed track, and MC for the true trajectory. In case (2), illustrated in figure

4.21a, the true muon trajectory passes through the plastic (MC top view) of the active plane,

around Z = 825 cm, deposits no energy in that plane, and then stops around Z = 970 cm.

The reconstructed track (Reco top view) ends in the previous active plane, which represents

a loss of about a third of the muon’s reconstructed energy. In case (3), illustrated in figure

78

(a)

(b)

(c)

Figure 4.21 Sample Events. Cases: (a) 2, (b) 3, and (c) (4). Only the top view of thedetector in each event is shown. The red tracks are reconstructed objects, and the bluetracks are the corresponding true objects. MC simulation.

79

4.21b, the tracker does not include in the muon track the cell hit left in the active plane

around Z = 825 cm, making the reconstructed red track around 3 m shorter than its true

length, and its energy about 600 MeV too low. In case (4), illustrated in figure 4.21c, the

true muon trajectory leaves the detector (MC top view), at Z = 1050 cm, before reaching

the next active plane at Z = 1075 cm. With this topology, the tracker incorrectly assumes

the muon stops in the detector, which results in an energy estimation that is too low. These

three cases are not properly represented by the correction applied to the track length in case

(1), nonetheless, these will be corrected by the method described above since, in data, it is

not possible to make a distinction from case (1).

Figure 4.22 Sample Event. From figure 4.20 with extra track length correction applied forcontainment purposes. MC simulation.

For tracks that actually leave the detector (not contained), the length correction presented

above is insufficient to obtain the correct energy. Therefore, tracks with the potential to leave

the detector will be removed from the sample. The correction presented above is applied to

80

the reconstructed track length of muons to better estimate, on average, the energy of the

muon sample. The lack of information on the real endpoints of the muon tracks requires

further manipulation of the track lengths in order to determine their containment. Only for

containment purposes, an extra line segment is added to the muon tracks, with endpoints

beyond Z > 600 cm, so that the new reconstructed track length reaches the next active

plane, as seen in figure 4.22. The magenta line segment goes from the last cell hit in the

track to the next active plane in the projected muon trajectory. If the new endpoint of the

extension (magenta line segment) is within the detector volume in both views, the track will

be labeled as contained, while the uncontained tracks will be removed from the sample.

4.4 Data Quality

Figure 4.23 Number Of Active Channels Per Good Run. The good Run numberincreases with time: 13067 = 10/29/2011, and 13782 = 04/30/2012.

A number of quality checks are performed to make a list of good data files to be used in

81

the physics analyses. A good data file, or SubRun, meets the following criteria:

• The first SubRun of a Run is longer than 30 min.

• The file is not a collection of empty events.

• The file has less than 10% of noise-only events.

• The number of cell hits per active channel per event (hace) in the file is:

0 < hace < 0.06.

• The average number of slices (ns) in the file is: 1 < ns < 6.

• The average slice duration (ts) in the file is: 0 < ts < 1500 ns.

• The file shows a record of synchronized DCMs.

Also, good data files must have at least 4400 active channels, as seen in figure 4.23.

In the midst of changing conditions of the NuMI beam, the reconstruction process yields

a total number of neutrino candidate events that correlates well with the integrated num-

ber of POT [104], as shown in figure 4.24. Although the proton beam intensity varied

throughout the data taking period, the beamline components did not change. There are

a few times during the data taking period where the number of candidate events found

jumped or dropped slightly in a small region of integrated POT. The candidate event rate:

2× 10−17 events/POT, correlates well with that expected on the basis of the neutrino beam

energy spectrum and the cross sections contained within the simulation program.

The DAQ reads out cell hits per event in 500 µs wide trigger time windows centered at

about the time of an expected neutrino arrival. The average time of the cell hits per physics

82

Figure 4.24 Number Of Neutrino Candidates As A Function Of The POT. Colorsrepresent various beam configurations [104].

Figure 4.25 Time Of Event Slice In The Data Of The NOνA Prototype Detector.Full DAQ window. Official NOνA NDOS figure [105]. The bin size is 1 µs.

83

slice is shown in figure 4.25 [105], where a physics slice is considered an event8. In the event

display, the trigger time window is represented by a histogram that starts at −50 µs and

ends at +550 µs. The data are read in blocks of 50 µs, and the boundaries of these blocks

generally do not match the boundary of the 500 µs time window, i.e. a portion of the first

and last blocks belongs to the 500 µs time window, and the other portion does not. Two full

blocks, one at the beginning and one at the end of the 500 µs time window, are presented in

each event display. The amount of hits recorded in these two blocks falls rapidly for times

away from the boundaries of the 500 µs time window. The neutrinos from the NuMI beam

are expected within the beam spill, 10 µs long, centered at 222 µs into the trigger time

window. A trigger time window containing a selected neutrino candidate event is shown

in figure 4.26. The red arrow points to the peak of cell hits that comes from the selected

neutrino candidate event at about 220 µs.

Figure 4.26 Sample Trigger Time Window. A selected neutrino candidate event withno cosmic ray background.

The out-of-time NDOS data, all from cosmic rays, which is defined by (refer to figure

8The word event used before to refer to collections of cell hits occurring within 500 µs isalso used to refer to a physics slice which is suppose to represent the cell hits of a neutrinointeraction.

84

4.25) the time intervals9:

t < +216.5µs, and t > +227.5µs, (4.8)

has a constant rate in the time intervals:

−3.5µs < t < +216.5µs, and + 227.5µs < t < +436.5µs. (4.9)

The recorded cosmic ray rate rapidly goes to zero for:

t < −3.5µs, and t > +436.5µs. (4.10)

The neutrino signal from the NuMI beam appears at about 222 µs in figure 4.25, as ex-

pected. As discussed above, the beam spill interval is 10 µs wide. However the NDOS

timing resolution is 0.5 µs, and consequently the in-time window is defined by:

+216.5µs < t < +227.5µs, (4.11)

a window that is 11 µs wide for this NDOS analysis.

85

Time (nsec)0 100000 200000 300000 400000

nse

c3

10

×S

lices

/ 2

0

50

100

150

200

Slice Time. NDOS DATA.

Figure 4.27 Time Of Physics Slices In The Data Of The Prototype Detector. Notethe in-time data at about 220 µs.

4.5 Cosmic Rays

As a consequence of its location in a surface building, NDOS records millions of cosmic rays.

A constant rate of cosmic rays is recorded in each trigger window for:

−3.5µs < t < +436.5µs. (4.12)

Therefore, to establish the level of cosmic background in the neutrino data, cosmic rays

within this time interval, 429 µs wide10, are considered. The time interval used to establish

the constant cosmic rate is shown in figure 4.27. The bin size is 2 µs, four times the NDOS

9These time intervals exclude the beam window.10Note that the width of 429 µs results from the exclusion of the in-time data, which is

11 µs wide.

86

timing resolution11. A straight line fit to the out-of-time data allows to determine that the

cosmic ray rate is flat within the shown time interval. The parameters of the fit are:

y-intercept: (114.9 ± 1.5) Slices/2 × 103ns,

slope:(

(5.5 × 10−7) ± (6.1 × 10−6))

Slices/2 × 103ns2. (4.13)

The slope is consistent with zero, and its variation over the range of the time window

is of 0.2%. The timing histogram representing the trigger window of an event with four

reconstructed cosmic rays is shown in figure 4.28. The peaks coming from their cells hits

appear at about: 55 µs, 90 µs, 290 µs, and 380 µs. Note that there is no peak of hits at the

NuMI trigger time, around 222 µs. The event display of this particular event with cosmic

rays is presented in figure 4.29.

Figure 4.28 Sample Trigger Time Window. Only cosmic rays in the event.

In order to reduce the cosmic ray background that appears inside the NuMI trigger

window, the angular distribution of the out-of-time cosmic tracks is compared to that from

the MC simulated beam tracks. Three angles are studied: cos θX , cos θY , and cos θNuMI.

The first two angles are those between the unit vectors of the detector’s axes X and Y (i

11All the slices that passed the containment cuts discussed in section 5.2, and had areconstructed track with track length longer than 2 m are presented in figure 4.27. The timeassigned to a slice is an average of all its cell hits times.

87

NOvA - FNAL E929

Run: 13068 / 3Event: 138120 / NuMI

UTC Sat Oct 29, 201117:34:16.348061696 sec)µt (

0 100 200 300 400 500

hits

110

210

q (ADC)0 0.5 1 1.5 2 2.5 3 3.5 4

10×

hits

110

210

0 200 400 600 800 1000 1200 1400

x (

cm)

-100

-50

0

50

100

z (cm)0 200 400 600 800 1000 1200 1400

y (

cm)

-200

-150

-100

-50

0

50

100

150

200

Figure 4.29 Sample Cosmic Ray Event. This figure shows a sample cosmic ray eventwith four reconstructed cosmic rays.

88

Figure 4.30 Beam To Prototype Detector Coordinate Transformation Illustration.Graphical representation of the beam reference frame (blue) and the NDOS reference frame(black). The red arrow represents the direction of the neutrinos, from the NuMI beam, thatreach the NDOS. Not to scale.

and j), and the unit vector of the longest track in the event, vt:

i · vt = cos θX =Xf −Xi

L,

j · vt = cos θY =Yf − Yi

L,

k · vt = cos θZ =Zf − Zi

L, (4.14)

where Xi, Yi, and Zi are the starting point coordinates of the longest track in the event; Xf ,

Yf and Zf are the ending point coordinates of the track, and L is the track’s length. The

cos θNuMI is the cosine of the angle between vt and the unit vector of the neutrinos coming

89

from the NuMI beam, vN :

vN · vt = cos θNuMI = vxNv

xt + v

yNv

yt + vz

Nvzt ,

= vxN cos θX + v

yN cos θY + vz

N cos θZ ,

=dx

dscos θX +

dy

dscos θY +

dz

dscos θZ ,

=dz

ds

(

ds

dz

dx

dscos θX +

ds

dz

dy

dscos θY + cos θZ

)

,

=1dsdz

(

dx

dzcos θX +

dy

dzcos θY + cos θZ

)

. (4.15)

The origin of the NDOS reference frame is: dx′ = -0.29 m, dy′ = 92.21 m, and dz′ = 841.76 m,

with ds′ = 846.80 m, in the beam coordinates. The coordinates transformation derived in

[106] (see figure 4.30) allows to get that point in the NDOS reference frame: dx = -29.0 cm,

dy = 4300.8 cm, and dz = 78070.2 cm, with ds = 78188.6 cm. In the NDOS reference frame:

dsdz = 1.00151, dx

dz = 0.00037, and dydz = 0.055. From equation (4.15), the cos θNuMI is:

cos θNuMI =0.055 cos θY + cos θZ

1.00151. (4.16)

The ratio dxdz is assumed to be zero in the MC simulation as it is small compared to the other

fractions. The direction of vN is illustrated in figure 4.31.

Cosmic rays are a background to the neutrino signal. However the angle of a cosmic ray

muon is rarely parallel to the direction of the neutrino beam. The angular distribution of

the longest tracks in the simulated CC neutrino interactions are peaked in the direction of

the neutrino beam, as shown in figure 4.32. The cosmic rays and the simulated tracks from

neutrino events have peaks at: cos θX = 0, as shown in figure 4.32a, so that this variable does

not provide much discrimination power. Cosmic ray tracks have a strong preference for the

90

Figure 4.31 The Direction Of Neutrinos From The NuMI Beam In The NDOS.The direction of the neutrinos from the NuMI beam at the NDOS location is represented byred arrows drawn on top of the NDOS event display.

vertical direction, | cos θY | ≈ 1, while long tracks generated in neutrino interactions prefer

the horizontal direction, cos θY ∼ 0, as shown in figure 4.32b. The greatest discrimination

between cosmic rays and the longest tracks from neutrino interactions is in the angle with

respect to the beam direction. The neutrino interaction tracks have a peak at: cos θNuMI ≈ 1,

in contrast with the maximum exhibited by the cosmic rays at: cos θNuMI ≈ 0.1, as shown

in figure 4.32c.

Simple cuts in the angular distributions of the reconstructed tracks can make a clean

separation between neutrino data and the cosmic ray background, as will be discussed in

Chapter 5. All the reconstruction tools presented in this chapter aid to analyze the data

that passes the quality control checks summarized here.

91

XθCos -1 -0.5 0 0.5 1

PO

T)

20 1

0×

1.6

7 ×

sec

µ 1

1 ×

Trac

ks /

(0.0

2

0

10

20

30

40

. NDOS MC & Cosmic Data.XθLongest Track Cos

Reco MC

Cosmic

(a)Yθ Cos

-1 -0.5 0 0.5 1

PO

T)

20 1

0×

1.6

7 ×

sec

µ 1

1 ×

Trac

ks /

(0.0

2

0

20

40

60

80

100

120

. NDOS MC & Cosmic DATA.YθLongest Track Cos

Reco MC

Cosmic

(b)

NuMIθ Cos 0 0.2 0.4 0.6 0.8 1

PO

T)

20 1

0×

1.6

7 ×

sec

µ 1

1 ×

Trac

ks /

(0.0

2

0

20

40

60

80

100

120

. NDOS MC & Cosmic Data.NuMIθLongest Track Cos

Reco MC

Cosmic

(c)

Figure 4.32 Longest Track Angular Distributions. NDOS MC (red) and cosmic raydata (black). Longest track MC and out-of-time cosmic data (a) cos θX , (b) cos θY , and (c)cos θNuMI distributions.

92

Chapter 5

Event Selection

As discussed in section 4.4, an event is a set of data collected within a 500 µs time window;

however, many different physical processes can occur within this time. The purpose of event

selection is to identify probable neutrino interactions within this time window, and study

them in order to determine their nature, i.e. their interaction type (CC or NC), and the

neutrino type (νµ or νe). In the NDOS, the word event refers to a neutrino interaction

without confusion since the chances of two or more neutrino interactions in one 500 µs

window are extremely low.

Although muons are easy to identify in the NOνA detectors, energetic charged pions

and protons that behave like minimum ionizing particles (MIP) represent a background

to the muon sample1. When protons and charged pions undergo hard scattering, these

leave in the detector characteristic signatures, figures 5.1a and 5.1c are examples of these

signatures. These two simulated single particle events have momenta around 2 GeV/c, and

do not represent a background to the muon signal because they are either too short or too

heavily ionizing. In contrast, when protons and charged pions behave like a MIP, as seen

in figures 5.1b and 5.1d, these do represent a background to the muon signal. These two

simulated single particle events also have momenta around 2 GeV/c, however, these leave

different signatures in the detector, similar to that of muons. Hadrons with momenta around

1Electromagnetic showers also contribute, in smaller proportions, to the background.

93

(a) (b)

(c) (d)

(e)

Figure 5.1 Single Particle Events. (a) Proton: 1.9 GeV/c. (b) Proton: 1.9 GeV/c. (c)π+: 2.1 GeV/c. (d) π+: 1.9 GeV/c. (e) µ: 0.5 GeV/c. Colored dots represent the variousenergy depositions left in the detector by the particles. MC simulation.

94

2 GeV/c that behave like a MIP are not as much of a background to 2 GeV/c muons as

these are to 0.5 GeV/c muons, as seen in figure 5.1e.

Described in the sections to follow are the cuts applied to reject events without a final

state muon. The various efficiencies and purities of the selected sample are also presented

in the following sections. The cuts that remove cosmic rays from the selected event sample

are included as well.

5.1 Charged And Neutral Current Neutrino Interac-

tions

(a) (b)

Figure 5.2 Sample Events. (a) Sample NC event: νµ + p→ νµ +K+ +K0 + n+ π0. (b)

Sample CC event: νµ +12 C → µ+ p+ π− + π0. MC simulation.

CC neutrino interactions have a background that comes from NC neutrino interactions

which contain a hadronic track that mimics a muon. Two sample MC events are shown in

95

figure 5.2: one is a NC event (figure 5.2a), and the other one is a CC event (figure 5.2b). The

two events in figure 5.2 show a similar energy deposition, and a long MIP track. A longest

track with MIP behavior is shown in figure 5.2a; this particle is a K+ (red track). The event

also features a K0, a neutron and a π0. The neutrino energy is 5.0 GeV. In contrast, figure

5.2b has a muon as its longest track (green track). The event also has a π0 that decays into

two photons, a proton, and a π+. This is a DIS event. The neutrino energy is 1.9 GeV.

Figure 5.2a is an example of a NC MC simulated event that needs to be rejected in order to

achieve a clean CC sample. Section 5.3 is going to address the rejection criteria.


Eve

nts

0

5

10

15

20

Energy. Interaction Type. NDOS MC.νTrue

All

CC

NC

Figure 5.3 NDOS Neutrino Energy Distributions. Predicted neutrino energy distribu-tions discriminated by interaction type. All interactions (black), CC (blue), and NC (red)energy distributions. MC simulation.

From the MC simulation it is known that 69.4% of the neutrino interactions in the NDOS

are CC and 30.6% are NC, as seen in figure 5.3. The CCNC ratio rises rapidly with neutrino

energy, as shown in figure 5.4. For energies above 1.8 GeV the ratio is about 2.6, consistent

with measured cross sections ratios [44].

96


(CC

/ N

C)

Rat

io

0

1

2

3

True Energy. NDOS MC.νCC/NC Ratio vs.

Figure 5.4 MC Prediction Of The Ratio Of CC To NC Neutrino Interactions.Ratio as a function of neutrino energy for all simulated events.


CC

/ (C

C +

NC

)

0

0.2

0.4

0.6

0.8

1

True Energy. NDOS MC.νReconstructed CC Fraction vs.

Figure 5.5 Fraction of Reconstructed CC Events. Fraction of reconstructed CC eventsout of the total sample: CC + NC, with at least one reconstructed track. MC simulation.

97


Rec

onst

ruct

ed/T

rue

Rat

io

0

0.2

0.4

0.6

0.8

1

1.2 True Energy. Reconstructed/True Ratio. NDOS MC.ν

All

CC

NC

Figure 5.6 Ratio Of Reconstructed Over Simulated Events. Reconstruction ratio asa function of the predicted neutrino energy discriminated by interaction type. The ratio re-constructed/true for: all interactions (black), NC (red), and CC (blue) energy distributions.All reconstructed events must have at least one reconstructed track. MC simulation.

As presented in section 4.3.2.2, the requirement of a minimum of 8 hits to reconstruct a

3D track is a strong cut resulting in an efficiency that strongly favors the desired CC events,

as shown in figure 5.5. At neutrino energies less than 0.1 GeV, no events are reconstructed.

For energies above 0.1 GeV the CC sample with at least one reconstructed track is about

four times larger than the reconstructed NC sample. From the total number of reconstructed

events: 81.4% are CC and 18.6% are NC. Within this sample there are three ratios of interest:

all reconstructed events to all simulated events, all reconstructed CC events to all simulated

CC events, and all reconstructed NC events to all simulated NC events, as seen figure

5.6. Events with energies less than 0.1 GeV are not reconstructed by the reconstruction

algorithms due to the lack of information. The NC event reconstruction rate drops below

50% for event energy less than 1.8 GeV, and the CC reconstruction rate dramatically drops

98

for event energy below 0.4 GeV. Summarizing, 26.4% of all simulated interactions are not

reconstructed. The effects of the remaining NC background is discussed in section 5.3.

5.2 Event Containment Criteria

Reconstructed neutrino interactions are required to originate2 in the vertex region (VR)

defined as (refer to figure 5.7):

|X| < 106 cm,

|Y | < 172 cm,

288 cm < Z < 452 cm. (5.1)

These VR limits are chosen such that the number of neutrino interactions occurring in it

is maximum while the purity3 and efficiency4 of the limits are maximum as well. The VR

limits also ensure that full cell transverse areas are included.

The VR represents 8% of the total NDOS volume. The instrumented regions in front

(lower Z) and to the sides of the limits of the VR are used as veto regions in order to ensure

that the longest track of each event starts within the VR. Hits within these veto regions

imply that the track came from outside the VR. The reconstructed neutrino interaction 3D

coordinates, or vertex, are defined at the point with lowest Z coordinate of the longest track.

The simulation indicates that the containment of the reconstructed interaction vertex within

2See section 4.2.3See equation (5.2).4See equation (5.3).

99

Figure 5.7 Vertex Region Of The Prototype Detector.

the VR has a purity (p) of:

p =NR

NT= (96.7 ± 0.2)%, (5.2)

where NR is the number of reconstructed tracks starting within the VR that have a corre-

sponding true track starting within the VR, and NT is the total number of reconstructed

tracks starting within the VR. The efficiency (ǫ) is:

ǫ =NR

Ne= (95.6 ± 0.2)%, (5.3)

where Ne is the number of true tracks starting within the VR. The VR purity and efficiency

as a function of neutrino energy are summarized in table 5.1, at the end of this chapter.

To determine the energy of a muon it must range out and stop in the detector, therefore

it has to be contained within the detector. Containment of the longest track is also one of the

cuts designed to reduce background from cosmic ray muons. The longest track is required

100

Figure 5.8 Containment Region Of The Prototype Detector.

to be contained within a containment region (CR) defined as (refer to figure 5.8):

|X| < 119 cm,

|Y | < 184 cm,

288 cm < Z < 1406 cm. (5.4)

These CR limits are chosen such that the number of neutrino events contained in it is

maximum while the purity5 and efficiency6 of the limits are maximum as well. The CR

limits also ensure that full cell transverse areas are included. In order for a longest track

to be fully contained, this has to start within the VR and end within the CR. Outside the

CR boundaries there are three veto cells between the edges of the CR and the edges of the

detector7. Hits in these veto cells imply that the track is not fully contained, and therefore

5See equation (5.5).6See equation (5.6).7From figure 5.8 it is clear that between Z = 0 and Z = 288 cm there are more than

three cells. All the instrumented volume in this region is used as veto.

101

the particle left the detector; or it came from outside the detector. The simulation indicates

that the containment of the longest track within the CR has a purity (p) of:

p =NR

NT= (87.7 ± 0.4)%, (5.5)

where NR is the number of reconstructed tracks contained within the CR that have a corre-

sponding true track contained within the CR, and NT is the total number of reconstructed

tracks contained within the CR. The efficiency (ǫ) is:

ǫ =NR

Ne= (76.1 ± 0.5)%, (5.6)

where Ne is the number of true tracks contained within the CR. The CR purity and efficiency

as a function of neutrino energy are summarized in table 5.1, at the end of this chapter. These

purities and efficiencies are affected by the small number of active planes beyond the VR

that are used to identify if, and where, the muon stopped within the detector.

To verify the containment of a reconstructed longest track within the VR and the CR

the track must be 3D. The simulation indicates that the efficiency (ǫ) of reconstructed 3D

tracks is:

ǫ =NR

Ne= (79.6 ± 0.2)%, (5.7)

where NR is the number of reconstructed 3D tracks with an associated true track, and Ne

is the total number of true tracks. The efficiencies as a function of neutrino energy are

summarized in table 5.1, at the end of this chapter. The low number of active channels

outside the VR is responsible for the 20% of longest tracks that are only reconstructed as

2D tracks. In these cases, one of the views does not have enough information to reconstruct

102

a 2D track, therefore, with only one 2D reconstructed track, a 3D reconstructed track is

impossible to make.

5.3 Charged Current Event Selection

In order to identify the CC neutrino events from the NC and cosmic ray backgrounds,

the NDOS MC simulation is used find the physical variables that are most sensitive to

the differences between them. To make such distinctions, various reconstructed physical

variables are investigated.

5.3.1 Longest Track Length Cut

Track Length (cm)0 200 400 600 800 1000 1200

Tra

cks

0

200

400

600

Longest Track Length. NDOS MC.

Muon

Non-muon

Figure 5.9 Monte Carlo Simulated Longest Track Length Distributions For Muonsand Non-muons. Muon (blue), and non-muon (red) track length distributions. MC simu-lation.

103

The main characteristic of a νµ CC event of interest is that it has a muon. In these CC

events the longest track is typically that of the muon, and the longer the track, the lower is

the background from hadronic or electromagnetic tracks (non-muon), as seen in figure 5.9.

The next approach to identify CC events is to study the reconstructed longest track length

(LTL) for each event, and separately examine CC and NC samples. These tracks must start

within the VR and be 3D.

Track Length (cm)0 200 400 600 800 1000 1200

Tra

cks

0

100

200

300

Longest Track Length. NDOS MC. Contained.

All

CC

NC

Figure 5.10 Monte Carlo Simulation Of The Longest Track Length Distributions.All interactions (black), CC (blue), and NC (red) longest track length distributions. Alllongest tracks are contained. MC simulation.

The LTL distributions are shown in figure 5.10 for each interaction type. Based on these

distributions, a cut is chosen at:

LTL > 200 cm. (5.8)

Relation (5.8) divides the sample into two: the first one is that with LTL > 200 cm, which has

a 3.8% background coming from NC events, and the second one is that with LTL < 200 cm,

104

which has a 24.7% background coming from NC events. The contained reconstructed sample

shows that 63.9% of all the CC reconstructed tracks, and 17.1% of all the NC reconstructed

tracks are longer than 200 cm. The analysis sample has all the LTL > 200 cm and it is

assumed to be a CC sample with a 3.8% of NC background. The MC simulation indicates

that the purity (p) of the track length cut (TLC) is:

p =NR

NT= (94.5 ± 0.1)%, (5.9)

whereNR is the number of reconstructed tracks longer than 200 cm that have a corresponding

true track longer than 200 cm, and NT is the total number of reconstructed tracks longer

than 200 cm. The efficiency (ǫ) is:

ǫ =NR

Ne= (59.6 ± 0.3)%, (5.10)

where Ne is the number of true tracks longer than 200 cm. The TLC purity and efficiency as

a function of neutrino energy are summarized in table 5.1, at the end of this chapter. These

relatively low efficiencies come from the combined effect of the reconstructed 3D tracks, the

reconstructed end point of track, and contained tracks efficiencies.

5.3.2 Cosmic Ray Cuts

The containment cut on the longest track in an event removes a large fraction of the cosmic

rays that are within the NuMI time window. However, those that appear to stop within

the vertex region and slip through the containment cut can be mistaken for a CC neutrino

event.

105

(a) (b)

Figure 5.11 Cosmic Rays Rejection Zone By The Cut In cos θY . Reconstructed cosmictracks in red with (a) cos θY = 0.6, and with (b) cos θY = 1.0.

106

Additional cuts on the angles of the longest track, discussed in section 4.5, are necessary

to further reduce this background. In section 4.5 three cosines were defined. The cos θY

distributions for MC simulation and out-of-time data (see figure 4.32b) present clear differ-

ences between signal and background. The longest track cos θY ratio: background/signal,

becomes less than 1 for | cos θY | < 0.6. In order to reject cosmic rays, only in-time events

with longest tracks with | cos θY | < 0.6 are accepted. This cut rejects 90.7% of the cosmic

background, and keeps 94.4% of the neutrino data. The cos θNuMI distributions for MC

simulation and out-of-time data (see figure 4.32c) present a clear difference between signal

and background. The longest track cos θNuMI ratio: background/signal, becomes less than

1 for cos θNuMI > 0.6. To further reject cosmic rays, only in-time events with longest tracks

with cos θNuMI > 0.6 are accepted. This cut rejects 96.9% of the cosmic background, and

keeps 96.3% of the neutrino data. The two cuts combined reject: 98.5% of all cosmic rays,

and keep 93.4% of the neutrino data.

Two sample cosmic events that span the rejection region defined by the cut in cos θY are

shown in figure 5.11. A cosmic ray with cos θY = 0.6 is shown in figure 5.11a, and a cosmic

ray with cos θY = 1.0 is shown in figure 5.11b. Another two cosmic events that span the

rejection region, defined by the cut in cos θNuMI are presented in figure 5.12. A cosmic ray

with cos θNuMI = 0 is shown in figure 5.12a, and a cosmic ray with cos θNuMI = 0.6 is shown

in figure 5.12b.

To summarize, the cosmic cuts accept tracks that fulfill:

| cos θY | < 0.6, cos θNuMI > 0.6. (5.11)

The correlations between cos θY and cos θNuMI are shown in figure 5.13. Most of the cosmic

107

(a) (b)

Figure 5.12 Cosmic Rays Rejection Zone By The Cut In cos θNuMI. Reconstructedcosmic tracks in red with (a) cos θNuMI = 0, and with (b) cos θNuMI = 0.6.

NuMIθ Cos

0 0.2 0.4 0.6 0.8 1

Yθ C

os

-1

-0.5

0

0.5

1

0

100

200

300

400

500

. NDOS Cosmic DATA. NuMI

θ vs. Cos Y

θLongest Track: Cos

(a)NuMI

θ Cos 0 0.2 0.4 0.6 0.8 1

Yθ C

os

-1

-0.5

0

0.5

1

0

50

100

150

200

. NDOS MC. NuMI

θ vs. Cos Y

θLongest Track: Cos

(b)

Figure 5.13 cos θY vs. cos θNuMI. (a) Cosmic rays. (b) MC simulation. The orangerectangles define the acceptance region.

108

tracks occur in the rejection region defined above, as shown in figure 5.13a, while from figure

5.13b it is clear that the MC signal is mostly inside the acceptance region.

The out-of-time cosmic rays that pass the cuts in relations (5.11) are scaled to the size

of the trigger window8, and then subtracted from the in-time data that pass these cuts to

obtain the final νµ + νµ CC candidates sample. This subtraction is done bin by bin in the

number of selected neutrino candidates vs. neutrino energy figure9.

5.3.3 Minimum Ionizing Particle Cut

Track dE/dX (MeV/cm)0 2 4 6 8 10

Tra

cks

0

0.02

0.04

0.06

0.08

0.1

0.12

Track dE/dX. NDOS Cosmic DATA.Track dE/dX. NDOS Cosmic DATA.

Figure 5.14 Cosmic Track Mean dE/dX. NDOS cosmic data.

The longest track in each event is assumed to be a muon. However, a background of

other particles, mostly hadrons, is present. The reconstruction algorithms can mistake an

electromagnetic shower as a long single track. CC interactions of the νe component in the

8Recall relation (4.11).9See figure 8.6.

109

NuMI beam can produce an energetic electromagnetic shower. Most of the electromagnetic

tracks that fake a muon can be rejected on the basis on their energy deposition profile.

MIP Fraction0 0.2 0.4 0.6 0.8 1

0

20

40

60

80

-310×

Longest Track MIP Fraction. NDOS MC.

Areas Normalized

µ

µNon-

Figure 5.15 MIP Fraction. Muon (blue) and non-muon (red). Areas normalized. MCsimulation.

The best way to know the energy deposition profile for muons is to investigate cosmic

muons. The distribution of the mean dE/dX of contained cosmic muons is presented in

figure 5.14. From this figure it is possible to define a MIP fraction (MIPfrac) as the fraction

of cells within a track with:

1.1 MeV/cm ≤ dE/dX ≤ 2.7 MeV/cm, (5.12)

range that encompasses 90% of all the cosmic muons, which have a mean of 1.95 MeV/cm.

A MIP must have a MIP fraction close to 1. The MIP fraction for simulated muon and

non-muon tracks is shown in figure 5.15. The mean MIP fraction for muons is 0.75, while

that of non-muons is 0.65. 99.8% of all muons have MIP fraction higher than 0.4, while

110

69.8% of all non-muons have MIP fraction higher than 0.4. These percentages motivate a

new cut, the MIP cut, defined as:

MIPfrac > 0.4. (5.13)

The fractions of the distinct particles producing the longest reconstructed track (LRT) are

presented in figure 5.16. The fraction of events in which a muon produces the LRT is 0.85, the

other 0.15 is equally shared between protons, charged pions, electrons, and photons10. Some

of these protons and charged pions come from NC interactions, however most of them are

secondary particles from high y (defined in equation (2.17)) interactions in the CC sample.

Electrons come from the νe CC background to the νµ CC signal, and can be rejected based

on their energy deposition profile.

ParticleMuon Proton Pion Electron Photon Other

Fra

ctio

n

0

0.2

0.4

0.6

0.8

Longest Track Particle ID. NDOS MC.

Before MIP Cut

After MIP Cut

Figure 5.16 Longest Track Particle Identity. Fraction of particles prior (red) and pos-terior (blue) to the MIP cut. MC simulation.

Rejecting tracks based on relation (5.13) considerably reduces the fraction of electromag-

10Photons here refer to the primary particle that originates the electromagnetic shower.

111

Neutrino Typeµν

µν eν + eν

Fra

ctio

n

0

0.2

0.4

0.6

0.8

Neutrino Type. NDOS MC.

Before MIP Cut

After MIP Cut

(a)

Interaction TypeCC NC

Fra

ctio

n

-110

1

Interaction Type. NDOS MC.

Before MIP Cut

After MIP Cut

(b)

Figure 5.17 Consequences Of The MIP Cut. Before (red) and after (blue) the MIP cutfor (a) neutrino types, and (b) interaction types. MC simulation.

netic tracks (from νe + νe CC) in the νµ CC sample from about 4% to about 1%, as shown in

figure 5.16. The impact of this cut on other aspects of the analysis is minimal, as seen from

figure 5.17. The composition of the neutrino sample before and after the MIP cut is seen in

figure 5.17a, with 1.4% of νe + νe CC background. The cut reduces the NC background to

a 2.9% level from the initial 3.5%, as seen in figure 5.17b. The MC simulation indicates that

the purity (p) of the MIP cut is:

p =NR

NT= (91.3 ± 0.6)%, (5.14)

where NR is the number of reconstructed tracks with MIP fraction larger than 0.4 and

identified in true as a muon, and NT is the total number of reconstructed tracks with MIP

fraction larger than 0.4. The efficiency (ǫ) is:

ǫ =NR

Ne= (99.1 ± 0.2)%, (5.15)

where Ne is the number of true muons. The MIP cut purity and efficiency per energy bins

are summarized in table 5.1, at the end of this chapter.

112

After the event selection procedure is complete, the sought set of νµ CC events is obtained.

Each of these events goes through a process of energy reconstruction that produces the

neutrino energy distribution of the selected sample.

Table 5.1 shows that the efficiency on the track length cut (TLC) increases with neu-

trino energy as the LTL reconstruction efficiency increases with length. The purity on the

containment cut (CR) decreases with increasing neutrino energy as the muons become more

energetic and leave the detector more frequently. Table 5.1 also shows how the MIP cut pu-

rity decreases with increasing neutrino energy as the contained reconstructed longest track

is not a muon. This is because the energetic muon escapes the detector more frequently for

high y CC interactions. All the efficiencies and purities shown in table 5.1 are calculated

to know the performance of each particular selection criterion. The calculations done for a

particular selection criterion have no dependency on previous or posterior selection criteria.

113

Cut\E (GeV) 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-10.0

3DEfficiency 74.8% 79.9% 80.2% 80.6% 81.3% 81.4% 80.3% 80.7%

Error 0.3% 0.3% 0.2% 0.3% 0.5% 0.7% 0.9% 0.5%TLCPurity 95.7% 94.6% 95.1% 94.8% 93.4% 92.4% 90.7% 92.2%Error 0.2% 0.2% 0.1% 0.2% 0.4% 0.5% 0.7% 0.4%

Efficiency 38.8% 53.8% 59.3% 63.2% 69.2% 71.5% 76.0% 81.6%Error 0.4% 0.4% 0.3% 0.4% 0.7% 0.9% 1.0% 0.6%VR

Purity 97.4% 97.0% 96.6% 96.6% 96.4% 95.4% 94.2% 95.3%Error 0.3% 0.2% 0.2% 0.2% 0.4% 0.5% 0.7% 0.4%

Efficiency 97.0% 96.4% 95.6% 95.5% 94.9% 94.8% 93.0% 93.9%Error 0.4% 0.3% 0.2% 0.2% 0.5% 0.6% 0.8% 0.4%CR

Purity 97.7% 94.4% 88.2% 83.6% 86.4% 80.0% 71.6% 55.6%Error 0.4% 0.4% 0.4% 0.5% 0.8% 1.2% 1.6% 1.1%

Efficiency 71.4% 68.8% 72.1% 73.8% 84.4% 86.2% 80.6% 73.2%Error 1.1% 0.7% 0.5% 0.5% 0.9% 1.0% 1.4% 1.0%MIPPurity 99.1% 94.3% 90.0% 87.7% 87.2% 83.3% 77.7% 58.6%Error 0.3% 0.5% 0.6% 0.7% 1.4% 1.8% 2.6% 2.4%

Efficiency 99.9% 99.8% 99.9% 99.9% 99.6% 100% 99.4% 98.4%Error 0.1% 0.1% 0.1% 0.1% 0.2% 0 0.5% 0.6%

Table 5.1 Event Selection Performance. Purity and efficiency for the various cuts madeduring the event selection. The columns in this table represent the energy bins used inthis analysis. The last column includes energies from 4.0 GeV to 10 GeV. No statisticallysignificant data exist below 0.5 GeV.

114

Chapter 6

Event Energy Reconstruction

The energy of a neutrino interaction must be determined from the energy deposited in

the detector by the final state particles. In νµ CC events, different physical processes are

involved in the determination of the energy of the muon or the hadronic system. In some

cases, a muon will stop in the detector before it decays or is captured in an atomic orbit.

The ionization energy loss mechanism of a muon is well understood, so that the energy

of a muon can be directly related to its range, which is estimated by its track length in

the detector1. The energy deposited in the detector by hadrons is reconstructed using the

detector as a calorimeter, i.e. the deposited energy is scaled, using the hadronic energy in

the MC simulation, in order to estimate the full energy encompassed within the hadronic

system.

6.1 Muon Energy Estimation

The MC simulation of the particles interacting with the detector incorporates the ionization

properties of the muon as well as the distortions introduced by the detector structure and

operation. For muon energies between 0.4 GeV to above 3 GeV, and the mixture of materials

of the NOνA detectors (average detector density of 1.02 g/cm3 [107]), the corresponding

ionization energy loss rates vary from 2.1 MeV cm2/g to 2.3 MeV cm2/g. As the muon

1See [87] for a detailed discussion on passage of particles through matter.

115

Track Length (cm)0 500 1000 1500 2000

Ene

rgy

(GeV

)µ

0

2

4

6

0

10

20

30

: True Energy vs. Reconstructed Track Length. NDOS MC.µ

(a)

Track Length (cm)0 500 1000 1500 2000

Ene

rgy

(G

eV)

µM

ean

0

2

4

6

: True Energy vs. Reconstructed Track Length. NDOS MC. Profile.µ

(b)

Figure 6.1 Muon: True Energy vs. Reconstructed Track Length. (a) True energy ofmuons as a function of the reconstructed track length. (b) Profile plot of (a). MC simulation.

energy drops below 0.4 GeV, the energy loss rate rises and reaches higher ionization levels

shortly before stopping. Therefore, the experimentally determined muon energy and the

range are nearly linearly related. The true muon energy and its reconstructed track length

are strongly correlated, as shown in figure 6.1. The reconstructed track length is used to

determine the energy of each observed muon. The true energy of the muons (ET ) as a

function of their reconstructed track length (LR) is (refer to figure 6.1a):

ET = p0 + p1LR, (6.1)

where the pi are parameters to be determined. Deviations from equation (6.1), particularly

at short track lengths, come from uncontained muons, as described in section 4.3.2. Equation

(6.1) is a good approximation to describe the relationship between the true energy and the

reconstructed track length (longer than 200 cm) of the muons; however, the data shown in

figure 6.1a do not follow a straight line, these are scattered inside a narrow band with some

width. In order to reconstruct the energy of the muon, the most probable value of true energy

per reconstructed track length bin is calculated and drawn as a function of reconstructed

116

track length, as shown in figure 6.1b. The error bars are the spread of the energy per track

length bin.

The pi in equation (6.1) are functions of LR, and are set equal to parameters obtained

from straight line fits. More than one straight line fit are required to better represent different

regions in figure 6.1b. The region: LR < 580 cm (refer to figure 6.2a), has χ2 = 90.91 for

ndf = 74. The region: 580 cm < LR < 920 cm (refer to figure 6.2b), has χ2 = 72.34 for

ndf = 66. The last region: LR > 920 cm (refer to figure 6.2c), has χ2 = 2634.5 for ndf = 144,

and is more scattered than the first one. The three regions combined have χ2 = 9340.46 for

ndf = 284. In order to minimize the χ2, a handful of straight line fits are calculated for the

first and last regions. For the last region, e.g., there are five fits with the following results:

• 920 cm < LR < 1115 cm: χ2 = 56.6 for ndf = 37.

• 1115 cm < LR < 1285 cm: χ2 = 53.4 for ndf = 32.

• 1285 cm < LR < 1375 cm: χ2 = 30.2 for ndf = 16.

• 1375 cm < LR < 1490 cm: χ2 = 30.6 for ndf = 20.

• 1490 cm < LR: χ2 = 132.3 for ndf = 31.

All the straight line fits are constrained to be continuous. The parameters of the fits are

used to obtain a first order approximation of the reconstructed energy.

The final version of the reconstructed muon energy is obtained after taking the difference

of the true energy and the approximated reconstructed energy. Small corrections, Cb, of the

order of 1% of the muon energy, are added to these approximated energies, per track length

bin, in order to achieve that the mean of the difference between the true energy (ET ) and

the reconstructed energy (ER) of the muon is equal to zero: ∆E = ET −ER = 0, as seen in

117

figure 6.3a. Finally, the reconstructed energy of the muon is:

ER = p0(LR) + p1(LR)LR + Cb(LR). (6.2)

The parameters of the fits and corrections are used to obtain the reconstructed muon energy

in data.

(a) (b)

(c)

Figure 6.2 Profile Plots Of Figure 6.1a. (a) LR < 580 cm. (b) 580 cm < LR < 920 cm.(c) LR > 920 cm.

The uniformity of the density of the NDOS is broken at the muon catcher, thus, muons

that reach the muon catcher require an additional term in equation (6.2) that would take

into account the portion of the muon’s trajectory that goes through the steel. For these

118

muons, equation (6.2) changes to:

ER = p0(LR) + p1(Ls)Ls + p2(Lµc)Lµc + Cb(LR), (6.3)

where Ls is the portion of the muon’s trajectory that goes through the plastic + scintillator

bulk, the p2 are fit parameters, and Lµc is the portion of the trajectory that goes through

the muon catcher. The reconstructed track length is: LR = Ls + Lµc.

Energy (GeV)µ 0 1 2 3 4

E (

GeV

)∆

-4

-2

0

2

4

0

20

40

60

80

100

120

: Energy Resolution vs. Reconstructed Energy. NDOS MC. µ

(a)

Mean 0.009899RMS 0.2966Constant 726.7Mean -0.04572Sigma 0.1303

E (GeV)∆-4 -2 0 2 4

Tra

cks

0

200

400

600

800

E. NDOS MC. ∆ Energy Resolution: µ

(b)

Figure 6.3 Muon Energy Resolution. (a) True minus reconstructed muon energy (∆E)as a function of the reconstructed muon energy. The average ∆E is effectively zero. (b)Overall muon energy resolution. MC simulation.

The overall muon reconstructed energy resolution is obtained from the Gaussian fit shown

in figure 6.3b. The spread in the ∆E from the fit is the reconstructed energy resolution.

The tail of the distribution accounts for about 5% of the muons. These are considered

contained, but their true trajectory leaves the detector, therefore the difference between true

and reconstructed energies is positive and not close to zero. From the spread in the fit,

the resolution in reconstructed muon energy is 130 MeV. The muon energy resolution at

energies higher than 2 GeV is better than lower energies since the containment issues are

less frequent when compared to shorter, and less energetic, muon tracks. High energy muons

119

Energy (GeV)µ 0 1 2 3 4 5

Muo

ns /

0.2

GeV

310

0

0.5

1

1.5

2

2.5

Energy Spectra. NDOS MC.µTrue & Reconstructed

True MC

Reco MC

Figure 6.4 Muon Energy Distributions. Muon true (black), and reconstructed (red)energy distributions. MC simulation.

often have their momentum vector almost parallel to the beam’s direction, and these reach

the muon catcher, which helps to contain them. Around 1 GeV the resolution is about 14%,

i.e. 140 MeV.

Even with the best attempts to properly reconstruct the muon energy, a comparison of

the simulated energy for muons in CC events with their reconstructed values shows that there

is a tendency to move muons with energies lower than 0.6 GeV to an energy above 0.6 GeV,

as shown in figure 6.4. The reconstructed muons with energies about 1 GeV sharply peak

while the true muon energy is more spread out. This is the result of the LTL corrections

described in section 4.3.2.

120

6.2 Hadronic Energy Estimation

The NOνA detectors were designed to obtain excellent electromagnetic shower identification

and energy resolution. This is accomplished with a detector comprised of primarily carbon

and hydrogen (low Z nuclei), resulting in an electromagnetic sampling fraction better than

65% [107]. These characteristics of the detector, however, result in a poor performance as a

hadronic calorimeter.

CC interactions produce hadrons in addition to the characteristic lepton. Hadronic energy

estimation is different from electromagnetic energy estimation (see [87] for details). Recalling

section 4.3.2, protons and charged pions often do not leave hit patterns to be reconstructed

into tracks in the NOνA detector, resulting in degraded energy resolution [108]. The presence

of hadronic showers combined with the neutrino event containment details studied in section

4.3.2, have as a consequence that the hadronic energy estimation is done by converting

deposited energy in the detector to real deposited energy.

The observed hadronic energy is defined as the sum of all the deposited energy in the

event that does not belong to the longest track, which is assumed to be the muon. Different

zones in the NDOS VR measure deposited energy differently depending on their location

and number of active channels. Since the neutrino event containment is only determined

by the longest track properties, it is necessary to introduce a method to take into account

hadronic energy that leaves the detector, is deposited in dead material or in cells without

APDs. Therefore, the energy corrections are determined by dividing the NDOS VR into 6

zones, as shown in figure 6.5. The limits of these zones are defined by:

• Zone 1: 53 cm > |X| > 106 cm, 288 cm < z < 345 cm.

• Zone 2: 53 cm > |X| > 106 cm, 345 cm < z < 402 cm.

121

Figure 6.5 Hadronic Energy Classification Zones. The coordinates of the edges of thedistinct zones are shown with arrows. All zones fulfill the condition: 172 cm > |Y|. Zone 1is the union of the two zones labeled with the number: 1. These two zones are equivalent,and therefore are united into one. The same is true for the zones 2 and 3. Zones 4 through6 are in the middle of the detector.

• Zone 3: 53 cm > |X| > 106 cm, 402 cm < z < 452 cm.

• Zone 4: 53 cm < |X|, 288 cm < z < 345 cm.

• Zone 5: 53 cm < |X|, 345 cm < z < 402 cm.

• Zone 6: 53 cm < |X|, 402 cm < z < 452 cm.

All 6 zones are rectangular boxes fulfilling the condition: 172 cm > |Y|. Events are

classified by the zone in which their vertex is located. Events with an interaction occurring

in zones 1 through 3 will have a higher probability for energy to leave through the sides

of the detector, when compared to the other zones. In addition, zone 3 will have a higher

probability to leak energy into the region without active cells, when compared to the other

zones. Events with its neutrino interaction occurring in zone 6 will deposit a good fraction

of their energy into the region without active cells. Events with its neutrino interaction

122

occurring in zones 4 and 5 will deposit most of the hadronic energy inside the detector and

the NDOS VR. All these zones have inactive or missing channels. Due to the differences in

energy deposition mentioned above, each zone needs its own energy estimation parameters.

Energy estimation parameters for each zone are determined through a comparison between

the deposited hadronic energy (the energy read by detector) and the true energy of the

hadronic system.

Deposited Energy (GeV)0 0.5 1 1.5 2

Mea

n H

adro

nic

Ene

rgy

(GeV

)

0

2

4

6

8Hadronic Energy: True vs. Deposited. NDOS MC. Zone 3. Profile.

(a)

Deposited Energy (GeV)0 0.5 1 1.5 2 2.5

Mea

n H

adro

nic

Ene

rgy

(GeV

)

0

2

4

6

8Hadronic Energy: True vs. Deposited. NDOS MC. Zone 4. Profile.

(b)

Figure 6.6 True Hadronic Energy vs. Deposited Energy. Mean true hadronic energyper deposited energy bin as a function of the deposited hadronic energy of (a) zone 3, and(b) zone 4. MC simulation.

The mean true hadronic energy, per deposited hadronic energy bin, as a function of the

deposited hadronic energy of zone 3 is shown in figure 6.6a. Zone 3 is the sample of events

that deposits less energy into the detector, when compared to the other zones. Figure 6.6b

is the equivalent of figure 6.6a for zone 4, which is the zone that gets the largest hadronic

energy deposition of all 6 zones. The correlations between true hadronic energy (HT ) and

deposited hadronic energy (HR), which are distinctly non-linear in all the 6 zones, help

to define the reconstructed hadronic energy (Ehad). These correlations are presented by

polynomials. The degree (A) of the polynomial is a function of the zone. The coefficients of

123

the polynomials (qa) are used to obtain the Ehad:

Ehad =A∑

a=0

qaHaR, (6.4)

There is one equation (6.4) per zone.

Reconstruted Energy (GeV)0 1 2 3 4 5

E (

GeV

)∆

-4

-2

0

2

4

1

10

210

310

E vs. Reconstructed Energy. NDOS MC.∆Hadronic:

(a)

Mean 0.06786RMS 0.4268Constant 1174Mean -0.02424Sigma 0.1909

E (GeV)∆-4 -2 0 2 4

Eve

nts

0

0.5

1

1.5

2

310×E. NDOS MC.∆Hadronic Energy Resolution:

(b)

Figure 6.7 Hadronic Energy Resolution. (a) True minus reconstructed hadronic energy(∆E) as a function of the reconstructed hadronic energy. The average ∆E is effectively zero.(b) Overall hadronic energy resolution. MC simulation.

Similar to the muon case, each equation (6.4) requires small corrections, Cg, of the order

of 5% of the hadronic energy, per deposited hadronic energy bin, in order to achieve that the

mean of the difference of energies (∆E = HT - Ehad = 0) in the hadronic system is equal

to zero, as seen in figure 6.7a. Finally, the hadronic reconstructed energy, Ehad, is:

Ehad =A∑

a=0

qaHaR + Cg(HR). (6.5)

The hadronic energy resolution, in figure 6.7a, at 1 GeV is 330 MeV (33%), at 180 MeV is

0.5 GeV (36.5%), and it is 70 MeV at 0.2 GeV (35%). From the Gaussian fit shown in figure

6.7b, the overall hadronic energy resolution is 190 MeV. The tails account for about 5% of

124

Hadronic Energy (GeV)0 1 2 3 4 5

Muo

ns /

0.2

GeV

310

0

0.5

1

1.5

2

2.5

True & Reconstructed Hadronic Energy Spectra. NDOS MC.

True MC

Reco MC

Figure 6.8 Hadronic Energy Distributions. True hadronic (black) and reconstructed(red) energy distributions. MC simulation.

all events.

The distributions in reconstructed and true hadronic energies shown in figure 6.8 have

bins of 200 MeV. From figure 6.6 it is clear that the spread of the data at energies above

1 GeV introduces deviations from the fits, and therefore underestimation of the hadronic

energies larger than 1.6 GeV is seen in figure 6.8.

6.3 Quasi-elastic And Non-quasi-elastic Classification

Section 2.1 presented differences between QE and non-QE interactions. Besides those differ-

ences, the two samples also present different efficiencies in the reconstruction process. These

two different neutrino interaction samples are used separately to calculate e.g. neutrino

fluxes and cross sections. Among the differences presented in section 2.1 is the hadronic

energy deposition, which is in general higher for non-QE interactions. This difference in

125

Hadronic Energy (GeV)0 1 2 3 4

Eve

nts

0

0.5

1

1.5

True Hadronic Energy Spectra. NDOS MC.

QE

Non-QE

Figure 6.9 QE And Non-QE Hadronic Energy Distributions. True hadronic QE (blue)and non-QE (red) energy distributions. MC simulation.

hadronic energy deposition is presented in figure 6.9. The mean true hadronic energy for

the QE sample is 150 MeV, while the mean for the non-QE sample is 910 MeV. The QE

true hadronic energy falls rapidly after 200 MeV, and it is at this same energy that the non-

QE true hadronic energy starts to become important. Therefore, a separation is possible

following the criteria:

Ehad < 200 MeV → QE, Ehad > 200 MeV → non-QE. (6.6)

With this separation, 84.1% of all true QE events, and 0.4% of all true non-QE events have

true hadronic energy lower than 200 MeV. Relations (6.6) are applied to the reconstructed

hadronic energy to make the separation between the QE and the non-QE samples.

The comparisons between true and reconstructed muon and hadronic energy are pre-

sented in figure 6.10. In these figures the true MC distributions represent the QE and

126


Fra

ctio

n of

muo

ns /

0.2

GeV

0

0.05

0.1

0.15

0.2

Energy Spectra. NDOS MC. QE.µTrue & Reconstructed

Areas = 1

True MC

Reco MC

(a)


Fra

ctio

n of

muo

ns /

0.2

GeV

0

0.1

0.2

0.3

Energy Spectra. NDOS MC. Non-QE.µTrue & Reconstructed

Areas = 1

True MC

Reco MC

(b)

Hadronic Energy (GeV)0 0.2 0.4 0.6 0.8 1

Fra

ctio

n of

eve

nts

/ 0.2

GeV

0

0.2

0.4

0.6

0.8

1

True & Reconstructed Hadronic Energy Spectra. NDOS MC. QE.

Areas = 1

True MC

Reco MC

(c)


Fra

ctio

n of

eve

nts

/ 0.2

GeV

0

0.1

0.2

0.3

True & Reconstructed Hadronic Energy Spectra. NDOS MC. Non-QE.

Areas = 1

True MC

Reco MC

(d)

Figure 6.10 Muon And Hadronic Energy Distributions For The QE And Non-QEEvents. The energy distributions for the true (black) and reconstructed (red) samples, withmuon energy in (a) QE and (b) non-QE events, and hadronic energy in the (c) QE and (d)non-QE events. Areas normalized to 1 in order to compare the shapes of the distributions.MC simulation.

127

non-QE interactions as defined in the MC simulation. The reconstructed MC distributions

represent the interactions as defined by relations (6.6). The poor tracking coverage for muons

immediately downstream of the VR causes muon energies below 0.6 GeV to be systemati-

cally reconstructed to energies higher than the true values by 0.2 GeV (in the QE case), and

0.4 GeV (in the non-QE case), as shown in figures 6.10a and 6.10b, respectively. Apart from

this bias at low muon energy, the distributions are similar for muon energies above 1.0 GeV.

The reconstructed hadronic energy in the QE sample is forced by definition to be less than

0.2 GeV, as shown in figure 6.10c. The reconstructed hadronic energy in the non-QE sample

is shifted slightly toward energies lower than the truth, as shown in figure 6.10d, due to a

bias for large hadronic energies to be reconstructed slightly lower in energy.

Mean -0.01601RMS 0.2207Constant 602.1Mean -0.04917Sigma 0.1337

E (GeV)∆-4 -2 0 2 4

Eve

nts

0

200

400

600

E. NDOS MC, QE.∆ Energy Resolution: µ

(a)


E (GeV)∆-4 -2 0 2 4

Eve

nts

0

0.2

0.4

0.6

0.8

1

310×E. NDOS MC, Non-QE.∆ Energy Resolution: µ

(b)

Energy (GeV)µ 0 1 2 3 4

E (

GeV

)∆

-4

-2

0

2

4

0

20

40

60

80

: Energy Resolution vs. Reconstructed Energy. NDOS MC, QE.µ

(c)

Energy (GeV)µ0 1 2 3 4

E (

GeV

)∆

-4

-2

0

2

4

0

50

100

150

: Energy Resolution vs. Reconstructed Energy. NDOS MC, Non-QE.µ

(d)

Figure 6.11 Muon Energy Resolutions For QE And Non-QE Events. Overall (a) QEand (b) non-QE energy resolutions. Muon (c) QE and (d) non-QE true minus reconstructedenergy (∆E) as a function of muon reconstructed energy. The average ∆E are effectivelyzero. MC simulation.

128

Following the procedure presented in section 6.2 gives the muon and hadronic energy

resolutions. The muon energy resolution for the QE sample at 1 GeV is 150 MeV (15%)

and at 2 GeV is 100 MeV (5%). The overall muon energy resolution for the QE sample

is 130 MeV, as shown in figure 6.11a. The muon energy resolution for the non-QE sample

at 1 GeV is 150 MeV (15%), and at 2 GeV is 130 MeV (6.5%). The overall muon energy

resolution for the non-QE sample is 160 MeV, as shown in figure 6.11b. The true minus

reconstructed muon energies as a function of reconstructed muon energy are shown in figure

6.11c, for QE, and in figure 6.11d, non-QE events.

A comparison between figures 6.3a and 6.11d shows that the muon energy resolution

around 0.8 GeV comes from non-QE muons that escape the detector, but are reconstructed

as contained. This effect was discussed in section 4.3.2. These muons deviate from the

original neutrino direction due to the high y of the event.

The hadronic energy resolutions in the QE and non-QE samples are different, as shown

in figure 6.12. The overall hadronic energy resolution is: 60 MeV for the QE sample, as

shown in figure 6.12a, and 240 MeV for the non-QE sample, as shown in figure 6.12b. Both

hadronic energy resolutions, at the average energy deposition in each case, are 30%.


E (GeV)∆-3 -2 -1 0 1 2 3

Eve

nts

0

0.2

0.4

0.6

0.8

1

310×E. NDOS MC, QE.∆Hadronic Energy Resolution:

(a)


E (GeV)∆-4 -2 0 2 4

Eve

nts

0

200

400

600

800

E. NDOS MC, Non-QE.∆Hadronic Energy Resolution:

(b)

Figure 6.12 Overall Hadronic Energy Resolutions. (a) QE and (b) non-QE events. MCsimulation.

129

6.4 Neutrino Energy Estimation

Energy (GeV)ν 0 2 4 6

E (

GeV

) ∆

-3

-2

-1

0

1

2

3

0

100

200

300

E vs. Reconstruted Energy. NDOS MC.∆: ν

(a)

Energy (GeV)ν 0 2 4 6 8 10

E (

GeV

) ∆

-3

-2

-1

0

1

2

3

0

100

200

300

E vs. True Energy. NDOS MC.∆: ν

(b)


E (GeV)∆-4 -2 0 2 4

Eve

nts

0

0.2

0.4

0.6

0.8

1

1.2310×

E. NDOS MC.∆ Energy Resolution: ν

(c)

Figure 6.13 Neutrino Energy Resolution. Neutrino true minus reconstructed energy(∆E) as a function of the (a) reconstructed (the average ∆E is effectively zero) and (b) trueneutrino energy. (c) Overall neutrino energy resolution. MC simulation.

The reconstructed neutrino energy (Eν) in each event is the sum of the two previously

estimated energies:

Eν = ER + Ehad + Cu(ER + Ehad), (6.7)

where Cu are small corrections, of the order of 3% of the neutrino energy, per neutrino

energy bin, introduced in order to achieve the difference between true neutrino energy and

reconstructed neutrino energy centered at zero, as seen in figure 6.13a. At the peak of the

neutrino energy distribution, near 2.0 GeV, the neutrino energy resolution is 200 MeV (10%).

130

The neutrino energy resolution as a function of the true neutrino energy is presented in figure

6.13b, where the effect of uncontained muons reconstructed as contained is more evident for

energies higher than 3 GeV. The overall neutrino energy resolution is 255 MeV, as shown in

figure 6.13c. The tails account for about 5% of all events.

Energy (GeV)ν0 2 4 6

Eve

nts

/ 0.5

GeV

310

0

0.5

1

1.5

2

2.5

Energy Spectra. NDOS MC.νTrue & Reconstructed

True MC

Reco MC

Figure 6.14 Neutrino Energy Distributions. The neutrino true (black) and reconstructed(red) energy distributions. Both distributions have the same number of events. MC simula-tion.

The neutrino energy distributions, with bins of 500 MeV, are in reasonable agreement, as

shown in figure 6.14. The effect observed earlier of muons below 0.6 GeV being reconstructed

at higher energies is reflected in the movement of events with true neutrino energies between

0.5 GeV and 1.0 GeV to higher values in the reconstructed neutrino energy. Also, inefficien-

cies in the containment of muons account for the underestimation of the neutrino energies

for events with energies higher than 3 GeV. The tendency observed earlier to reconstruct

hadronic energies above 2.0 GeV to slightly lower values contributes as well. Those events

are reconstructed between the 1 GeV and the 2 GeV energy bins.

131

Energy (GeV)ν 0 1 2 3 4

E (

GeV

) ∆

-3

-2

-1

0

1

2

3

0

20

40

60

80

100

120

E vs. Reconstruted Energy. NDOS MC, QE.∆: ν

(a)

Energy (GeV)ν 0 1 2 3 4

E (

GeV

) ∆

-3

-2

-1

0

1

2

3

0

50

100

150

E vs. True Energy. NDOS MC. QE.∆: ν

(b)


E (

GeV

) ∆

-4

-2

0

2

4

0

50

100

150

200

E vs. Reconstruted Energy. NDOS MC, Non-QE.∆: ν

(c)


E (

GeV

) ∆

-3

-2

-1

0

1

2

3

0

50

100

150

200

E vs. True Energy. NDOS MC. Non-QE.∆: ν

(d)

Figure 6.15 QE And Non-QE True Minus Reconstructed Neutrino Energies. QEtrue minus reconstructed neutrino energy (∆E) as a function of (a) reconstructed and (b)true neutrino energy. Non-QE ∆E as a function of (c) reconstructed and (d) true neutrinoenergy. MC simulation.

132

From the separation of the neutrino sample between QE and non-QE events done in sec-

tion 6.3, the neutrino energy resolution, as a function of the reconstructed neutrino energy,

are as shown in figures 6.15a and 6.15c for the QE and non-QE samples respectively. The

neutrino energy resolution at 2 GeV is: 130 GeV (6.5%) for the QE sample, and 220 GeV

(10.1%) for the non-QE sample. The neutrino energy resolution as a function of the true

neutrino energy are shown in figures 6.15b and 6.15d for the QE and non-QE samples re-

spectively. From these figures, the effect of uncontained muons reconstructed as contained

is evident in the non-QE. These muons come from high y events, and their angles deviate

from the original neutrino angle, therefore the muons escape the detector before stopping.


E (GeV)∆-4 -2 0 2 4

Eve

nts

0

100

200

300

400

500

E. NDOS MC, QE.∆ Energy Resolution: ν

(a)


E (GeV)∆-4 -2 0 2 4

Eve

nts

0

200

400

600

E. NDOS MC, Non-QE.∆ Energy Resolution: ν

(b)

Figure 6.16 QE And Non-QE Neutrino Energy Resolutions. Overall (a) QE and (b)non-QE neutrino energy resolution. MC simulation.

The overall neutrino energy resolution for the QE sample is 170 MeV, as shown in figure

6.16a; and for the non-QE sample it is 300 MeV, as shown in figure 6.16b. The tails in

figure 6.16 account for about 5% of the events. The three neutrino energy resolutions, as a

function of the reconstructed neutrino energy, are compared in figure 6.17.

The neutrino energy distributions for QE and non-QE are presented in figure 6.18. In

these figures the true MC distributions represent the QE and non-QE interactions as defined

in the MC simulation. The reconstructed MC distributions represent the interactions as

133


Ene

rgy

Res

olut

ion

(GeV

)

0

0.1

0.2

0.3

0.4: Energy Resolution vs. Reconstructed Energy. NDOS MC.ν

All

QE

Non-QE

Figure 6.17 Comparison Of Neutrino Energy Resolutions. The neutrino energy reso-lutions for all (black), QE (blue), and non-QE events. MC simulation.


Fra

ctio

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eve

nts

/ 0.5

GeV

0

0.1

0.2

0.3

Energy Spectra. NDOS MC. QE.νTrue & Reconstructed

Areas = 1

True MC

Reco MC

(a)


Fra

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/ 0.5

GeV

0

0.1

0.2

0.3

Energy Spectra. NDOS MC. Non-QE.νTrue & Reconstructed

Areas = 1

True MC

Reco MC

(b)

Figure 6.18 QE And Non-QE Neutrino Energy Distributions. (a) QE and (b) non-QEneutrino true (black) and reconstructed (red) energy distributions. Areas normalized to 1 inorder to compare the shapes of the distributions. MC simulation.

134

defined by relations (6.6). In the QE and non-QE interactions, the area normalized true and

reconstructed neutrino energy distributions are in reasonable agreement. The distributions

for the QE and non-QE are distinct. The QE, with energy mean 28.4% smaller than that

of the non-QE, slowly varies between 0.5 GeV and 2.5 GeV, as shown in figure 6.18a. The

non-QE distribution is strongly peaked near 2.0 GeV, as shown in figure 6.18b. These effects

are the consequence of the cross sections dependence on the neutrino energy, as was shown

in figure 2.4. The two cross sections are equal (for νµ) at about 1 GeV; at this point QE is

reaching a plateau and non-QE is increasing. The event rate in figure 6.18 is dominated by

non-QE for energies higher than 1 GeV.

Neutrino Typeµν

µν eν + eν

Fra

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0

0.2

0.4

0.6

0.8

Neutrino Type. NDOS MC. QE & Non-QE.

QE

Non-QE

(a)


Eve

nts

0

0.5

1

1.5

True Energy Spectra. NDOS MC. QE & Non-QE.µν & µν

QEµν QEµν Non-QEµν Non-QEµν

(b)

Figure 6.19 Neutrino Types: Fractions And Energy Distributions. (a) Fraction ofneutrino types discriminated by the interaction type. (b) Neutrino energy distributionsdiscriminated by interaction type and neutrino type. No scale shown in order to comparethe shapes. MC simulation.

Each of the two interaction types is the sum of νµ and νµ components. Figure 6.19

presents this composition. The overall fraction of the different neutrino types discriminated

by interaction type is presented in figure 6.19a. The non-QE fraction is higher for interactions

generated by a νµ, and the QE fraction is higher2 for interactions generated by a νµ. The

neutrino energy distributions per neutrino type and interaction type are shown in figure

2This is true because νµ interacts with Hydrogen nuclei in QE interactions and the νµdoes not.

135

6.19b. The νµ contribution at the 2 GeV peak is about 10%. This contribution almost

triples at adjacent lower and higher energy bins from the peak, as a result of the various

cross sections that originate the interactions.

136

Chapter 7

Systematic Uncertainties

Systematic uncertainties are often related to measuring devices or measuring methods used

in experiments. A measuring device, or a detector, could consistently read the value of a

measured quantity with an offset from its real value. This offset could be due to an scale

factor, a calibration issue, or the materials and geometry of the detector. A measuring

method could under or over estimate a physical quantity due to the way it is designed or

implemented, which would result in a measurement that is consistently different from the

real value of the measured quantity.

There are six relevant systematic uncertainties on the number of events identified during

this analysis. The energy estimation of the neutrino candidate event relies on the absolute

energy calibration of the detector. Differences in the calibrations between data and MC

simulated events result in systematic uncertainties in the number of selected events per

energy bin. The various channels configurations of the detector during the period in which

the data was taken introduce systematic uncertainties in the number of selected events since

different configurations could disagree on whether a particular event is contained or not in

the detector. To minimize this uncertainty, each channels configuration in the data sample

was simulated in the MC sample. The uncertainty in the cross sections and the modeled final

state physics influence the number of selected events. In the case of the cross sections, the

measured number of events is directly proportional to the cross sections; and the model used

137

to describe the physics of the final state could increase or decrease the rate at which particular

types of events occur under the circumstances of the experiment. The mass of the target

region of the detector is also directly proportional to the measured number of events, and

therefore its uncertainty creates a systematic uncertainty on the number of selected events.

A mathematical procedure known as unfolding (see appendix C) shifts selected events from

the energy bins in which these are measured to the most probable energy bin for a particular

event. The shift is based on relations between true and reconstructed events that are built

into the algorithm. Since the rearrangement of events introduces a change in the number of

selected events per energy bin, the unfolding method also introduces a systematic uncertainty

on the number of selected events that is related to the parameters of the algorithm that are

subject of variation due to optimization criteria. Finally, the counting method of the POT

directly affects the relative number of selected events. The same absolute number of selected

events changes its interpretation depending on the reported POT. Thus, uncertainties in the

POT introduce systematic uncertainties in the number of selected events. Each one of these

major systematic uncertainties is presented in the following sections.

7.1 Energy Estimation Uncertainty

There are two key features in the energy estimation process: the length of the longest track

per event, and the deposited hadronic energy. To determine the systematic uncertainty of

the energy estimation of neutrino events in the NDOS, the mean energy deposited per unit

length in cosmic tracks are compared to MC simulated muon tracks, as seen in figure 7.1.

Differences in the two distributions imply a systematic difference in the estimated energy

of measured and simulated neutrinos. The two means of these distributions differ by 9.1%,

138

Track dE/dX (MeV/cm)0 2 4 6 8 10

-1T

rack

s / 0

.1 M

eV c

m

0

20

40

60

80

100

120

Logest Track dE/dX. NDOS MC & Cosmic DATA.

MC Area Normalized

Reco MC

Cosmic

Figure 7.1 Mean Energy Deposition Per Unit Length. dE/dX of cosmic (black) andreconstructed MC simulated tracks (red). The MC simulation area is normalized to data.

which is interpreted as a 10% systematic uncertainty on the deposited energy.

This 10% systematic uncertainty is added to and subtracted from the deposited energy in

equation (6.5) to find how the hadronic energy estimation changes. These changes affect the

MIP fraction of the tracks as well, which is taken into account when evaluating the change

in the number of events. Finally, these systematic uncertainty affects the QE and non-QE

separation since it is carried out using the estimated energy of the hadronic system.

The muon energy estimation comes from the measured track length using equation (6.2).

The comparison between the longest track lengths of reconstructed MC simulated and data

tracks is presented in figure 7.2. The mean of the MC simulated track length distribution is

9.6% larger than that of data. This difference is assumed as a 10% systematic uncertainty on

the track length measurement. This 10% systematic uncertainty is added to and subtracted

from the track length in equation (6.2) to find how the muon energy estimation changes.

139

Track Length (cm)200 400 600 800 1000 1200

PO

T)

20 1

0×

1.6

7 ×

sec

µT

rack

s / (

11

0

10

20

30

40

Track Length. NDOS MC & DATA.

Reco MC

Data

Figure 7.2 Comparison Of Data And MC Simulated Longest Track Length Distri-butions. Reconstructed MC simulated (red) and data (black) tracks.

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

erta

inty

(%

)

-10

-5

0

5

10

Energy Estimation Systematic Uncertainty. NDOS MC.

Figure 7.3 Energy Estimation Systematic Uncertainty As A Function Of The Neu-trino Energy.

140

These changes combined with those presented above, for the deposited energy, give the

systematic uncertainties1 to the number of events shown in figure 7.3. The impact of these

variations (δE) is of the order of 8%.

7.2 Prototype Detector’s Channels Configuration Un-

certainty

The NDOS active channels configuration varied over time, since APDs were added, removed,

and shifted during the period when the data were taken. To estimate the systematic uncer-

tainties introduced to the number of events due to these changes in the channels configura-

tion, three different configurations are examined (refer to figure 4.23): run numbers smaller

than 13250, run numbers bigger than 13250 and smaller than 13600, and run numbers bigger

than 13600. From the first set, the channels configuration of run 13220 is chosen because it

has the least number of active channels. From the second set, the channels configuration of

run 13258 is chosen because it is the one with the most active channels. Finally, from the

third set, the channels configuration of run 13703 is chosen because it has the most active

channels.

A set of MC simulated events is reconstructed with each of the three channels configura-

tions, and the number of reconstructed events is examined as a function of the run number.

The systematic uncertainties2 on the number of events due to the impact of the various chan-

nels configurations are shown in figure 7.4. The same event reconstructed under different

channels configurations could be selected as contained in some of them, and not selected in

1Figure 7.3 presents the uncertainties as percentages of the total number of events.2Figure 7.4 presents the uncertainties as percentages of the total number of events.

141

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

erta

inty

(%

)

-20

-15

-10

-5

0

Channel Configuration Systematic Uncertainty. NDOS MC.

Figure 7.4 Channels Configurations Systematic Uncertainty As A Function Of TheNeutrino Energy.

others because it was not reconstructed as contained. The lack of instrumented cells in key

points of the detector is responsible for this issue. The variation in the number of selected

events introduced by each channels configuration is compared to the nominal one, and the

maximum variation is used as the systematic uncertainty. The impact of these variations

(δC) is of the order of 15%.

7.3 GENIE Cross Sections And Final State Physics

Uncertainty

In order to give the users the ability to verify the validity of new models explaining nuclear

interactions, or study how possible variations to the current models could affect the physics

output of the simulations, GENIE has features that allow the users to introduce variations to

142

key parameters that will result in changes to the simulated outputs produced by the software.

The developers provided a mechanism to assign weights to simulated neutrino interactions,

which is implemented by the ReWeight package. The main concept behind this process is

that for each input physics quantity P , with uncertainty3 δP , which is taken into account,

there is a parameter xP such that:

P → P ′ = P

(

1 + xPδP

P

)

. (7.1)

P could be a single physical parameter, a simple function, or a prediction done by a MC

simulation. Some of these quantities may not be easy to write analytically or tabulated,

therefore it is preferable to formulate the problems in terms of xP . The parameter xP is an

integer, positive and negative values are allowed. When xP = 0, there would be no variation

to P . xP = ±1 indicates that P would change by one standard deviation, δP [109].

The reweighting package is named according to the final effect that it produces: it assigns

weights to individual MC simulated events based on the changes introduced by the xP . The

tool is very useful since it simplifies the analysis. Instead of running the MC simulation

as many times as changes are introduced to the models, the weights reflect the importance

of a given event on the sought final result. A weight equal to 1 means that the variation

in a given xP had no effect in the final result. A weight greater than 1 indicates that the

variation in xP produces an increase in the frequency of such weighted events proportional

to the size of the weight. A weight less than 1 means that the variation in xP produces

a decrease in the frequency of such weighted events proportional to the size of the weight.

The Mccqea parameter (defined in table 7.1) is used as an example of the ReWeight package

3δP is the estimated standard deviation of P .

143

Energy (GeV)ν 0 1 2 3 4 5

Eve

nts

0

200

400

600

GENIE ReWeight. NDOS MC.accqe

M

σ+Originalσ-

Figure 7.5 Example Of The ReWeight Package Output. The black distribution is theoriginal, unaltered, MC sample. The +1σ variation to M

ccqea is shown in red, and the -1σ

variation in blue.

output, which is presented in figure 7.5. The original neutrino energy spectrum features a MC

simulated sample of events that passed all the event selection cuts. The ReWeight package

assigned weights to each of the events in the original distribution, and the consequences

of such weights are seen in red for the +1σ variation, and in blue for the -1σ variation.

Although the number of entries in the three distributions is the same, the integrals in the

reweighted distributions change as a result of the applied weights by 7.2% for +σ, and by

6.0% for −σ. The percentage of variation in the number of events is used as the systematic

uncertainty for this particular parameter. This procedure is the same for all the studied

parameters.

The dominant parameters that affect this analysis, include the axial mass for CC QE

scattering, the nuclear Pauli suppression effects in CC QE reactions4, the non-resonance

4Modifying the Fermi momentum is the way to affect the Pauli suppression.

144

xP Description of P δP/P

νµ interaction cross section systematic parameters

Mccqea Axial mass for CC QE +25% -15%

Mccresa Axial mass for CC resonance neutrino production ±20%

Rνncc1πbkg Non-resonance bkg in νn CC 1π reactions ±50%

Hadronization systematic parameters

fz Hadron formation zone ±50%Intranuclear hadron transport systematic parameters

MFPπ π mean free path (total rescattering probability) ±20%MFPn Nucleon mean free path (total rescattering probability) ±20%Frninel Nucleon inelastic reaction probability ±40%Cexn Nucleon charge exchange probability ±50%Absn Nucleon absorption probability ±20%Frπ Nucleon π production probability ±20%

Table 7.1 GENIE Parameters. Taken from [109].

background for all CC 1π and 2π production channels, CC DIS, intranuclear hadron transport

model, and total rescattering probability for hadrons within the target nucleus5. Some of

these categories have more than one parameter available for the reweighting process. All the

parameters tabulated in tables 8.1, 8.2, and 8.3 of [109] are studied in this analysis, however

not all of them introduce finite changes to the number of events. All the parameters that

introduced finite variations to the number of events are presented in table 7.1, which also

shows their standard deviations.

There are three categories in table 7.1: neutrino interaction cross section systematic

parameters, intranuclear hadron transport systematic parameters, and hadronization sys-

tematic parameters. Hadrons produced in the nuclear environments are not obtained from

a single interaction, initially quarks propagate through the nucleus with low probability of

interaction as these are not yet part of a hadron. GENIE models this effect by defining

5This systematic is related to the mean free path of the hadron within the nucleus.

145

Energy (GeV)ν1.5 2 2.5 3 3.5

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erta

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)

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-10

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10

20GENIE Parameters Systematic Uncertainty. NDOS MC.

accqeMaccresMbkg

πncc1νR

inelnFr

πMFPnMFP

nCexnAbs

πFrfz

Figure 7.6 GENIE Parameters Systematic Uncertainty As A Function Of TheNeutrino Energy. The parameters shown are defined in table 7.1.

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

erta

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(%

)

-20

-10

0

10

20

Total GENIE Systematic Uncertainty. NDOS MC.

Figure 7.7 GENIE Parameters Total Systematic Uncertainty As A Function OfThe Neutrino Energy. The sum of the parameters defined in table 7.1 is done in quadra-ture.

146

a free step, fz = pcτ0/m, where p is the hadron’s momentum, τ0 = 0.342 fm/c is the

formation time, m is the mass of the hadron, and c is the speed of light. The formation zone

is defined by the distance between the intranuclear event vertex and the recorder position of

the hadron at the beginning of the intranuclear cascade step [109].

The systematic uncertainties6 in the number of events obtained for the parameters pre-

sented in table 7.1 are summarized in figure 7.6. The impact of the combined variations of

all the studied parameters (δG) is of the order of 15%, in the energy region where most of the

data occurs, i.e. 1.5 GeV to 2.5 GeV. The sum of these uncertainties is done in quadrature,

and is shown in figure 7.7.

7.4 Unfolding Systematic Uncertainty

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

erta

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(%

)

-5

0

5

Unfolding Systematic Uncertainty. NDOS MC.

Figure 7.8 Unfolding Algorithm Systematic Uncertainty As A Function Of TheNeutrino Energy.

6Figure 7.6 presents the uncertainties as percentages of the total number of events.

147

The unfolding procedure presented in appendix C uses two parameters that can modify

the output of the algorithm, hence the number of events would vary accordingly. These

parameters are: k, which sets the number of statistically significant equations, and nTSVD,

which is the number of toy MC simulated events generated by the TSVDUnfold tool to

calculate the uncertainties on the unfolded distribution. To find the systematic uncertainty

in the unfolded distribution of the number of events, the k parameter is modified to be: 4

and 6, the original value was 2; and the nTSVD is modified to be: 50 and 200, the original

value was 100. The systematic uncertainties7 in the number of events obtained after varying

these two parameters are shown in figure 7.8. The impact of these variations (δU) is of the

order of 5%.

7.5 Total Systematic Uncertainty

Before discussing the total systematic uncertainties, there three more systematic uncertain-

ties to be considered in this analysis: that related to the POT calculation, that related to the

fiducial mass, and that related to the fraction of neutrinos resulting from each charged meson

decay, i.e., the charged kaons to pions ratio, in the number of selected neutrino candidates.

NOνA uses the same POT counting scheme used by MINOS, thus the 2% systematic uncer-

tainty (δP ) used in their study of νµ disappearance [110] is assumed in this analysis. The

fiducial mass has a 5% systematic uncertainty which comes from the comparison between

the physical measurements performed on the materials of the detector and the parameters

extracted from the MC simulation.

A. Lebedev [111] measured the ratio of charged pions and kaons production by 120 GeV/c

7Figure 7.8 presents the uncertainties as percentages of the total number of events.

148

protons incident on a carbon target. The production ratios were measured in longitudinal

momenta from 20 GeV/c to 90 GeV/c. These are not the longitudinal momenta of the mesons

involved in the NDOS measurement, however, the lack of measurements in the longitudinal

momentum region concerning this analysis (around 7 GeV/c for kaons) leads to use the

measurements in [111] as a constraint on the K/π ratio at the momentum region of interest.

The ratios in [111] are in agreement with the ratios embedded in FLUKA within 10%. This

value is used as the systematic uncertainty on the number of neutrinos coming from charged

pion decays (δR) in this analysis.

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

erta

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(%

)

-20

0

20

40Summary of Systematic Uncertainties. NDOS MC.

GENIE ParametersCh. Config.Energy Est.

/KπPOTFiducial Mass

Figure 7.9 Summary Of Systematic Uncertainties. The systematic uncertainties: δG(magenta), δC (dark blue), δE (red), δR (green), δP (light blue), and δm (grey), are pre-sented as a function of the neutrino energy.

The total systematic uncertainty associated to the number of events combines all the

systematic uncertainties discussed in the previous sections. The systematic uncertainties

presented in figure 7.3, 7.4, and 7.6, for energy estimation (δE), channels configuration

(δC), and GENIE parameters (δG), respectively, are combined along with the systematic

149

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

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-1

-0.5

0

0.5

1

Number of Events Systematic Uncertainty. NDOS DATA.

QEµν Non-QEµν QEµν Non-QEµν

Figure 7.10 Systematic Uncertainty On The Number Of Events As A Function OfThe Neutrino Energy. Each neutrino kind and interaction type is represented by a color.

uncertainty associated to the POT (δP ), the fiducial mass (δm), and the charged kaons to

pions ratio (δR). All these uncertainties are dimensionless, and represent the uncertainty

in the number of events due to each of the variables mentioned above. The correlations

between all these uncertainties are of the order of the effect themselves, hence these second

order effects are ignored, and the uncertainties are considered independent of each other,

therefore these can be combined in quadrature:

δN =√

δE2 + δC2 + δG2 + δP 2 + δm2 + δR2. (7.2)

The summary of all these uncertainties, which contribute to obtain the uncertainty in the

number of selected events (δN ) before the unfolding procedure is applied, is shown in figure

7.9. There are four equations (7.2) associated with each energy bin, one for each interaction

type and neutrino type: νµ QE, νµ non-QE, νµ QE, and νµ non-QE.

150

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

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-1

-0.5

0

0.5

1N/N Ratio. NDOS DATA.δ

Systematics

Statistics

Figure 7.11 δN/N As A Function Of The Neutrino Energy. Ratios between the sys-tematic uncertainty and the measured number of events (red), and the statistical uncertaintyand the measured number of events (blue).

The combination of all the systematic uncertainties given by equation 7.2 are presented

in figure 7.10. Once equation 7.2 is computed, the unfolding algorithm is applied, and a new

unfolded uncertainty in the number of selected events (δN ′) is generated. The total system-

atic uncertainty on the unfolded number of events (δN) is the combination in quadrature of

δN ′ and δU :

δN =√

δN ′2 + δU2. (7.3)

The ratios between the total systematic uncertainties, δN , and the measured number of

events, and the statistical uncertainties and the measured number of events, are compared

in figure 7.11. Statistical and systematic uncertainties are of the same order, about 12%, in

the region of the charged kaons peak in the neutrino energy, i.e. from 1.5 GeV to 2.5 GeV.

151

Chapter 8

Analysis

The event selection criteria presented in chapter 5, and the energy reconstruction process

presented in chapter 6, are applied to the NDOS data in order to find the observed number

of νµ + νµ CC events.

8.1 Data And Monte Carlo Simulation Comparison

YθCos -1 -0.5 0 0.5 1

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120

. NDOS DATA.YθLongest Track Cos

In-time

Out-of-time

(a)NuMIθ Cos

0 0.2 0.4 0.6 0.8 1

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80

. NDOS DATA.NuMIθLongest Track Cos

In-time

Out-of-time

(b)

Figure 8.1 Angles Of The Longest Track. (a) Longest track’s cos θY for out-of-timecosmic data (blue), and in-time data (black). (b) Longest track’s cos θNuMI.

NDOS data are separated (see section 4.4) into those recorded in-time (within the 11 µs

NuMI time window), where mixtures of neutrino interactions and cosmic rays are expected,

and out-of-time, where only cosmic rays are expected, as shown in figure 8.1. The cosmic

ray background expected in the in-time window is the out-of-time data normalized by the

152

ratio of the time windows widths, given by:

Cosmic → Cosmic × 11

429. (8.1)

The regions where cosmic rays are the dominant contribution are: | cos θY | > 0.6, and

cos θNuMI < 0.6. The in-time distributions without a cosmic ray background, obtained by

subtracting the out-of-time data, can be compared with the distributions predicted by the

MC simulation, as shown in figures 8.2a and 8.2b. In these figures, all the cuts presented in

chapter 5 (see equations (5.1, 5.4, 5.8, and 5.13)) are applied, except the angular cuts. The

MC simulation distributions are normalized to the data POT such that:

MC → MC × POTData

POTMC= MC × 1.67386 × 1020

7.32109 × 1021. (8.2)

The background subtracted data has the general characteristics predicted by the MC

simulation with the cos θY distribution peaked near zero, and the cos θNuMI distribution

strongly peaked near 1. The only significant deviation is that the number of events predicted

by the MC simulation is higher than is seen in data, which is evident in the reduced height

of the peaks at: 1 in the cos θNuMI, and 0 in the cos θY distributions. When the shapes

of the distributions are compared, these are in good agreement, as shown in figures 8.2c

and 8.2d. Figure 8.2 illustrates the effect of eliminating the out-of-time data background.

However, in the analysis, the cosmic background subtraction of the data is done at the

number of events level (see figure 8.6). In order to verify that the subtraction of out-of-time

data from in-time data was adequate, five random 11 µs time windows from the out-of-time

data sample are chosen and treated as in-time data. The remaining out-of-time data are

153

Yθ Cos -1 -0.5 0 0.5 1

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. NDOS MC & DATA.YθLongest Track Cos

MC

Data

(a)NuMIθ Cos

0 0.2 0.4 0.6 0.8 1

PO

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sec

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100

. NDOS MC & DATA.NuMIθLongest Track Cos

MC

Data

(b)

YθCos -1 -0.5 0 0.5 1

PO

T)

20 1

0×

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sec

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1 ×

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cks

/ (0.

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. NDOS MC & DATA.YθLongest Track Cos

MC

DataMC Area Normalized

(c)NuMIθ Cos

0 0.2 0.4 0.6 0.8 1

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. NDOS MC & DATA.NuMIθLongest Track Cos

MC Area Normalized

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(e)NuMIθCos

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sec)

µ 1

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cks

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02

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. NDOS Cosmic DATA.NuMIθLongest Track Cos

(f)

Figure 8.2 Longest Track Angular Distributions Without Cosmic Background. Theangular distributions for the longest track with a subtraction of the cosmic ray backgrounddetermined from out-of-time data. Longest track’s cos θY MC simulation (red) and data(black) (a) POT normalized and (c) MC simulation area normalized to data. Longest track’scos θNuMI (b) POT normalized and (d) MC simulation area normalized to data. Data means:in-time data minus out-of-time data. Longest track’s (e) cos θY and (f) cos θNuMI, for fakein-time data minus normalized out-of-time data.

154

normalized as indicated in equation (8.1), however this time the factor of 429 is replaced by

a factor of 418 to take into account that now two, and not just one, 11 µs time windows are

subtracted from the full 440 µs time window. The out-of-time data are subtracted from the

fake in-time data, and the results are consistent with zero, as seen in figures 8.2e and 8.2f.

Each of the five fake in-time data samples chosen from the out-of-time data present similar

behaviors to the ones seen in these two figures. The deviations from zero seen in figure 8.2a,

for | cos θY | > 0.6, and in figure 8.2b, for cos θNuMI < 0.5, are consistent with the deviations

(of statistical origin) seen in figure 8.2e and 8.2f, respectively, for the same intervals. These

fluctuations occur in the regions where most of the cosmic data exists, however, the data in

these regions is rejected by the cosmic cuts presented in equation (5.11), hence, these do not

affect the analysis.


Muo

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Figure 8.3 Reconstructed Muon Energy Distributions, No Cosmic Background.Reconstructed muon energy in the MC simulation (red) and in-time data (black) for (a) QEand (b) non-QE interactions with a subtraction of the cosmic ray background determinedfrom out-of-time data. MC simulation area normalized to data.

The neutrino energy is reconstructed by adding the calorimetric energy and the muon

energy from range. These individual components in the data are compared to the MC

simulation, however, given the discrepancy in normalization between the MC simulation and

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the recorded data described above, only shape comparisons of the individual components are

made. In addition, a cosmic background subtraction has been applied. The MC simulation

provides a good representation for the shape of the muon energy distribution in the non-QE

sample, as shown in figure 8.3b. The QE distributions require a deeper analysis because

there are several data points below the MC near 2 GeV. A Kolmogorov-Smirnov test [112]

is done in order to verify if the prediction of the MC simulation agrees with the data. The

test results give a maximum deviation (D) of: D = 0.0233, a Kolmogorov-Smirnov test

statistic (KS) of: KS = 0.628, and a p-value of: p = 0.825. Based on these results, the

two QE distributions are in good agreement. A p-value greater than 0.1 implies that there

is a low probability for the null hypothesis to be false. In this case, the null hypothesis is

that the two cumulative distribution functions (CDF) in figure 8.4 are drawn from the same

distribution, i.e. these represent the same physics. A CDF describes the probability that a

random variable (sample value) R, with a given probability distribution F (R), will be found

to have a value less than or equal to r [112].

The QE and non-QE distributions in figure 8.3 share the peak around 1 GeV. The muon

energy distribution for QE interactions exhibits another peak in energy just below 2 GeV.

This reflects the maximum in the neutrino flux at 2 GeV due to charged kaon decays, and

the general lack of energy in the hadronic system of a QE neutrino interaction. These 2 GeV

muons are likely to be contained only if these have cos θNuMI ≈ 1, otherwise these are likely

to leave the detector. There are more contained 2 GeV muons in the QE sample because

these are likely to keep the direction of the neutrino, therefore traveling towards the muon

catcher. Non-QE events have higher y, hence their muons are, in general, less energetic.

The simulated hadronic energy distribution in the non-QE sample is clearly consistent

with the recorded data, as shown in figure 8.5. This will enable a reliable neutrino energy

156

Figure 8.4 Cumulative Distribution Functions For Muons. Data (red) and MC simu-lation (green).


Eve

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MC Area Normalized

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Figure 8.5 Hadronic Energy Distributions, No Cosmic Background. ReconstructedMC simulation (red) and in-time data (black) non-QE energy distributions. MC simulationarea normalized to data. Cosmic background subtracted.

157

calculation using the sum of the muon and hadronic energies. The mean hadronic energy is

around 0.7 GeV, which is about half of the mean energy of the muons.

Event Energy (GeV)0 1 2 3 4 5

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In-time

Out-of-time

Figure 8.6 Selected Candidate Events Energy Distributions. In-time (black) andout-of-time (blue) data energy distributions with cosmic cuts applied.

The level of the cosmic ray background remaining in the in-time data is obtained by ap-

plying all cuts to the out-of-time data. Near 1 GeV this cosmic ray background contribution

is approximately 10% of the number of events seen in the in-time data, as shown in figure

8.6. The consequences of the cuts, presented in table 5.1 and in equation (5.11) on the data

sample are summarized in table 8.1. 239 νµ + νµ CC candidate events are selected. The

cosmic background subtraction is done to this set of selected candidate events.

The number of selected QE candidate events (Mq) is the sum of a number of true QE

events (Tq) and a number of true non-QE events, or fake QE events (Fq), as seen in the first

row of the matrix in figure 8.7. Similarly, the number of selected non-QE candidate events

(Mn) is the sum of the rest of the true non-QE events (Tn) and the rest of the true QE

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Cut Name Number of Events

All events (no cut) 1460853D 93722LTL 54986VR 3715

Containment 993MIP 939

Cosmic 239

Table 8.1 Event Selection Cuts. Summary of the effect of all the event selection cuts onthe data sample. The number of events represents the number of selected neutrino candidateevents that passed each cut. Cosmic background subtraction is applied to the events set thatpassed all these cuts.

(p, ǫ)\E (GeV) 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-10.0

QEPurity 97.4% 89.1% 86.2% 84.0% 91.2% 87.4% 50.0% 85.7%Error 0.5% 0.7% 0.6% 0.7% 1.0% 1.3% 2.7% 1.6%

Efficiency 78.4% 71.7% 71.6% 68.1% 64.4% 48.1% 22.0% 35.3%Error 1.2% 0.9% 0.7% 0.8% 1.6% 2.0% 2.3% 2.2%

Non-QEPurity 34.6% 79.1% 88.1% 87.5% 85.8% 82.0% 89.8% 97.7%Error 1.4% 0.8% 0.5% 0.6% 1.2% 1.5% 1.7% 0.7%

Efficiency 84.5% 92.4% 94.8% 94.5% 97.2% 97.2% 96.9% 99.8%Error 1.0% 0.6% 0.4% 0.4% 0.6% 0.7% 1.0% 0.2%

Table 8.2 Quasi-elastic And Non-quasi-elastic Classification Performance. Puritiesand efficiencies for QE and non-QE sample. The columns represent neutrino energy bins.

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Figure 8.7 Efficiency And Purity Matrix For The QE And Non-QE Samples. Thismatrix aids to construct the efficiencies and purities related to the selection of reconstructedQE and non-QE candidate events.

events, or fake non-QE events (Fn), as seen in the second row of the matrix in figure 8.7:

Mq = Tq + Fq, Mn = Tn + Fn. (8.3)

The efficiencies (ǫq and ǫn) and purities (pq and pn) for the selection of QE and non-QE

candidate events are defined as:

ǫq =Tq

Nq, ǫn =

Tn

Nn,

pq =Tq

Mq, pn =

Tn

Mn, (8.4)

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where the total number of real QE events (Nq) is:

Nq = Tq + Fn = Mqpq +Mn(1 − pn), (8.5)

and the total number of real non-QE events (Nn) is:

Nn = Tn + Fq = Mnpn +Mq(1 − pq). (8.6)

The purity (see equation (8.4)) of the QE sample is (89.1 ± 0.8)%, and (84.4 ± 0.4)% for

the non-QE sample. The efficiency (see equation (8.4)) of the QE sample is (71.2 ± 1.0)%,

and (94.5 ± 0.3)% for the non-QE sample. The QE and non-QE samples purity and efficiency

per energy bins are summarized in table 8.2. Once the number of selected neutrino candidate

events (from the NDOS data), for QE and non-QE, is measured, the efficiencies and purities

in table 8.2 are used to obtain the real number of selected candidate events per interaction,

and per energy bin.

Three sample MC simulated events to illustrate the kind of topology that the selection,

presented in relation (6.6), allows in the QE sample are shown in figure 8.8. The event

display in figure 8.8a represents the following interaction:

νµ(0.8 GeV/c) +12 C → µ(0.7 GeV/c) + p(0.2 GeV/c). (8.7)

The proton does not leave the interaction cell, and the muon’s track is about 3 m long. There

is no track reconstruction shown in these figures, and the colors of the cell hits represent the

amount of energy deposited, which follows the color code on the lower right histogram. The

reconstructed neutrino energy for this event is: 1.15 GeV, and the reconstructed hadronic

161

(a)

(b)

(c)

Figure 8.8 Sample Of Simulated QE Neutrino Events.

162

energy is: 0.1 GeV. The interaction in the event display shown in figure 8.8b is:

νµ(1.9 GeV/c) +12 C → µ(1.8 GeV/c) + p(0.2 GeV/c) + 2n(0.3 GeV/c). (8.8)

The proton does not leave the interaction cell, and the muon’s track is about 7 m long. The

two neutrons do not deposit any reconstructable energy. This is the typical νµ CC event

expected to be found in the NDOS data, at the 2 GeV peak. The reconstructed neutrino

energy for this event is: 1.90 GeV, and the reconstructed hadronic energy is: 0.1 GeV. The

event display in figure 8.8c represents the following interaction:

νµ(3.0 GeV/c) +12 C → µ(2.8 GeV/c) + 4p+ n(0.3 GeV/c). (8.9)

There are four protons, each with about 0.2 GeV/c, that deposit all their energy in the

interaction cell, and a neutron. The muon has 2.8 GeV/c, which takes it half way into

the muon catcher. Muons of these energies are only likely to be contained if these reach

the muon catcher. The reconstructed neutrino energy for this event is: 2.85 GeV, and the

reconstructed hadronic energy is: 0.1 GeV. The three sample MC simulated events are good

representations of the QE sample, these deposit very little energy in cells that do not belong

to the reconstructed track.

In contrast, three sample events selected by relation (6.6) illustrate the kind of topologies

allowed in the non-QE sample. The event display in figure 8.9a represents the following

interaction:

νµ(2.0 GeV/c) + p→ µ(1.3 GeV/c) + p(0.4 GeV/c) + π+(0.6 GeV/c). (8.10)

163

(a)

(b)

(c)

Figure 8.9 Sample Of Simulated Non-QE Neutrino Events.

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The proton does not leave the interaction cell, the muon’s track is about 6 m long, and the

π+ leaves a MIP-like track with a hard scattering at the end. This is a resonance event. The


energy is: 0.9 GeV. The event display in figure 8.9b represents the following interaction:

νµ(1.8 GeV/c) +12 C → µ(0.8 GeV/c) + p(0.9 GeV/c) + π+(0.6 GeV/c). (8.11)

Although this interaction is similar to the previous one, in this case the proton leaves a track

with high energy depositions (top view, top track), and the π+ leaves a shower with some

electromagnetic energy deposition associated with it (top view, bottom track). The muon’s

track is about 3 m long. This is a resonance event. The reconstructed neutrino energy

for this event is: 1.95 GeV, and the reconstructed hadronic energy is: 0.8 GeV. The event

display in figure 8.9c represents the following interaction:

νµ(2.1 GeV/c)+35 Cl → µ(0.6 GeV/c)+p(0.3 GeV/c)+n+π+(0.7 GeV/c)+π+(0.4 GeV/c).

(8.12)

The proton deposits most of its energy in the cell around: Y = 20 cm and Z = 375 cm,

the 0.7 GeV/c π+ leaves a MIP-like track with a hard scattering at the end, the 0.4 GeV/c

π+ leaves a scattered shower going downwards in the side view, the neutron deposits no

reconstructable energy, and the muon’s track is about 2 m long. This is a DIS event. The


energy is: 1.55 GeV. The muons in the non-QE sample have energies around 1 GeV, and are

likely to be contained.

A sample of selected QE neutrino candidates, similar to the one presented in figure 8.8, is

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(a)

(b)

(c)

Figure 8.10 Sample Of QE Neutrino Candidates. (a) Selected neutrino candidate with1.0 GeV of estimated energy. (b) Selected neutrino candidate with 2.5 GeV of estimatedenergy. (c) Selected neutrino candidate with 3.3 GeV of estimated energy.

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(a)

(b)

(c)

Figure 8.11 Sample Of Non-QE Neutrino Candidates. (a) Selected neutrino candidatewith 1.9 GeV of estimated energy. (b) Selected neutrino candidate with 1.8 GeV of estimatedenergy. (c) Selected neutrino candidate with 2.0 GeV of estimated energy.

167

shown in figure 8.10. The color of the cell hits represent times, and these follow the color code

of the timing histogram in the bottom left of figure 8.10. Figure 8.10a presents a selected QE

neutrino candidate with 1.0 GeV of reconstructed energy, similar to that presented in figure

8.8a. Figure 8.10b presents a selected QE neutrino candidate with 2.5 GeV of reconstructed

energy, similar to that presented in figure 8.8b. Figure 8.10c presents a selected QE neutrino

candidate with 3.3 GeV of reconstructed energy, similar to that presented in figure 8.8c.

Another sample of selected non-QE neutrino candidates, similar to the one presented in

figure 8.9, is shown in figure 8.11. Figure 8.11a presents a selected non-QE neutrino candidate

with 1.9 GeV of reconstructed energy, similar to that presented in figure 8.9a. Figure 8.11b

presents a selected non-QE neutrino candidate with 1.8 GeV of reconstructed energy, similar

to that presented in figure 8.9b. Figure 8.11c presents a selected non-QE neutrino candidate

with 2.0 GeV of reconstructed energy, similar to that presented in figure 8.9c. These six

selected neutrino candidates share similar topologies to the MC simulated ones, hence, MC

simulated and data events are likely to share the same type of neutrino interaction, as well

as the same final state particles. The full list of selected neutrino candidates is presented in

appendix D.

After the subtraction of the cosmic ray background, and the separation of the QE and

non-QE samples, the number of neutrino events predicted by the MC simulation is higher,

in each case, by about 30% than is observed in the data, as shown in figure 8.12. This is

agreement with the observation that the MC simulation predicted more muon tracks than

appeared in the data. When the MC simulated distribution is normalized to the number of

data events, the shapes are quite similar, as shown in figure 8.13.

The distributions presented in figure 8.12 are reconstructed MC simulation and data.

There are differences between reconstructed and true MC simulated distributions, as pre-

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Figure 8.12 Number Of Neutrino Candidates vs. Neutrino Energy, No CosmicBackground. Number of selected neutrino candidate events vs. neutrino energy for (a) QEand (b) non-QE interactions. Reconstructed MC simulation (red) and in-time data (black)energy distributions with a cosmic background subtraction and normalized by POT.


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Figure 8.13 Number Of Neutrino Candidates vs. Neutrino Energy, No CosmicBackground, Shape Comparison. (a) QE and (b) non-QE number of neutrino eventsvs. neutrino energy. Reconstructed MC simulation (red) and in-time data (black) energydistributions. MC simulation areas normalized to data.

169

sented in section 6.4, that can only be resolved by the unfolding procedure that accounts

for the inefficiencies and resolutions of the detector, and provides the transformation of the

reconstructed event energy into corrected energy bins.


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Figure 8.14 QE And Non-QE Unfolded Number Of Neutrino Candidates vs. Neu-trino Energy, No Cosmic Background. (a) QE and (b) non-QE number of selectedneutrino candidate events vs. neutrino energy after unfolding. True MC simulation (green)and unfolded data (black) energy distributions normalized by POT.

The unfolding algorithm is applied to the data distributions1 in figures 8.12a and 8.12b,

and the results are shown in figure 8.14. The unfolded distributions should be compared to

the true MC simulated distributions. There are 32.4% more QE predicted events by the MC

simulation than unfolded candidate events. There are 35.9% more non-QE predicted events

by the MC simulation than unfolded candidate events. These differences in the predicted and

candidate events come, thus, from an extra factor of 1/3 in the normalizations included in the

MC simulation. The shapes of the data and MC simulated distributions are in agreement,

as see in figure 8.15, where the MC simulation areas are normalized to data.

1Figures 8.15 should be compared with figures 8.13 to appreciated the effect of the un-folding procedure.

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Figure 8.15 QE And Non-QE Unfolded Number Of Neutrino Candidates vs. Neu-trino Energy, No Cosmic Background, Shape Comparison. (a) QE and (b) non-QEnumber of neutrino events vs. neutrino energy after unfolding. True MC simulation (green)and unfolded data (black) energy distributions. MC simulation areas normalized to data.

8.2 Results

The inclusive CC event sample is obtained by combining the QE and non-QE neutrino energy

distributions presented in figure 8.14. The total number of selected νµ + νµ CC candidate

events is: 229.7 per 1.67 × 1020 POT. This number is not an integer because it comes from

the subtraction of out-of-time data from in-time data. The number of events in the in-time

data set is an integer. The out-of-time data set is scaled to the size of the in-time data time

window (11 µs), and therefore is not necessarily an integer. The sums of the two interactions,

which are the reconstructed MC simulation and the data (without unfolding), are shown in

figure 8.16a. With the MC simulation normalized to the number of data events, the shapes of

the inclusive sample are in good agreement, as seen in figure 8.16b. The true MC simulation,

with 328.1 predicted events per 1.67 × 1020 POT, exhibits 30% more predicted events than

the unfolded candidate events, as shown in figure 8.17a. Comparing the shapes between the

true MC simulated and unfolded data distributions, between 0.5 GeV and 4 GeV in figure

8.17b, indicates that there is a good agreement among them. The agreement is corroborated

171

by the χ2 test, which in this case concludes that χ2 = 5.549 for ndf = 6.


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Figure 8.16 Number Of Neutrino Candidates vs. Neutrino Energy, Comparison OfMC Simulation And Data. Energy dependence of the reconstructed MC simulation (red)and data without unfolding (black) normalized by (a) POT, and (b) with MC simulationareas normalized to data.


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Figure 8.17 Unfolded Number Of Neutrino Candidates vs. Neutrino Energy, Com-parison Of MC Simulation And Data. True MC simulation (green) and unfolded data(black) normalized by (a) POT, and (b) with MC simulation areas normalized to data.

The number of selected CC candidate events per energy bin (NbinCC) in figure 8.16 has

no cosmic background2, 2.9% of overall NC background, and 1.4% of overall νe + νe CC

background (see section 6.1). Table 8.5, at the end of the chapter, summarizes the percent-

2Due to cosmic background subtraction.

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ages of NC background per energy bin, and table 8.6, at the end of the chapter, summarizes

the percentages of νe + νe CC background per energy bin. These two tables feature the

backgrounds present in the NbinCC .

ν kind\E (GeV) 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0

νµQE: Mνµq 16.8 13.2 14.9 11.2 2.3 1.3 0.2

νµ Non-QE: Mνµn 1.8 19.8 39.0 34.1 8.4 5.2 1.6

νµQE: Mνµq 8.9 6.5 3.4 2.5 1.6 1.0 0.1

νµ Non-QE: Mνµn 1.2 5.5 4.7 3.7 3.2 1.9 0.6

NbinCC 28.8 45.0 62.0 51.5 15.4 9.4 2.5

Table 8.3 Candidate Events With Background. Number of selected candidate eventsper neutrino type and interaction type. The cosmic background was subtracted, thoughthese numbers still reflect the NC and νe + νe CC background presence. The last row, Nbin

CC ,is the total number of selected candidate events per energy bin. See equation (8.3) for thedefinition of the Mν

q and Mνn .

The total number of selected candidate events, NT , is the sum of the CC QE (Nq) and

CC non-QE (Nn) neutrino candidates (see equations (8.5) and (8.6)), and each of these two

numbers is the sum of the νµ and νµ contributions:

NT = Nνµq +N

νµn +N

νµq +N

νµn . (8.13)

After the background subtraction is applied, the Mνq and Mν

n in table 8.3 are substituted

into equations (8.5, 8.6) yielding Nq and Nn respectively.

The final results, the number of selected νµ + νµ CC candidate events per energy bin

(NbinT ), are summarized in table 8.4. Statistical and systematic uncertainties are of the

same order, with the systematic error typically larger. The total uncertainty is the sum in

quadrature of the two types of uncertainties, as presented in the full discussion on systematic

uncertainties in chapter 7. These NbinCC are about 2/3 of the prediction done by the MC

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ν Energy (GeV) Candidates/1.67×1020 POT stat. error sys. error total error

0.5 to 1.0 28.28 4.47 +7.60−1.38

+8.82−4.68

1.0 to 1.5 42.30 6.50 +5.83−2.43

+8.73−6.94

1.5 to 2.0 58.54 6.79 +8.91−5.82

+11.2−8.94

2.0 to 2.5 48.56 6.42 +6.62−6.75

+9.22−9.32

2.5 to 3.0 14.06 2.29 +4.13−1.81

+4.74−2.96

3.0 to 3.5 8.63 1.98 +4.66−1.75

+5.07−2.64

3.5 to 4.0 2.31 1.06 +1.08−0.35

+1.51−1.11

Table 8.4 Number Of Candidate Events. Selected νµ + νµ CC candidates per neutrinoenergy bin. Statistical and systematic errors are shown in independent columns. The totalerror combining the statistical and systematic errors are shown in the last column.

simulation. Chapter 9 addresses the possible causes of the disagreement between NDOS

data and the MC simulation.

ν kind\E (GeV) 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0

νµ QE 0.1% 0 0.9% 0.5% 1.1% 1.1% 0%Uncertainty ±0.1% 0 ±0.9% ±0.5% ±1.0% ±1.0% 0%νµ Non-QE 0 0.6% 1.7% 3.9% 4.0% 6.4% 7.1%Uncertainty 0 ±0.6% ±1.3% ±2.0% ±2.1% ±2.5% ±2.7%νµ Non-QE 0 0.1% 0.1% 0.2% 1.0% 0.5% 1.1%Uncertainty 0 ±0.1% ±0.1% ±0.2% ±1.0% ±0.5% ±1.0%νe Non-QE 0 0 0.1% 0.1% 0.3% 0.2% 0.4%Uncertainty 0 0 ±0.1% ±0.1% ±0.3% ±0.2% ±0.4%

Table 8.5 Neutral Current Background. The NC background per energy bin as a functionof the interaction type. The rows in this table represent the interaction type.

174

ν kind\E (GeV) 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0

νe + νeQE 1.6% 3.0% 2.4% 0.9% 0.5% 1.1% 0Uncertainty ±1.3% ±1.7% ±1.5% 0±.9% ±0.5% ±1.0% 0

νe + νe Non-QE 0.8% 1.2% 0.6% 0.5% 1.7% 1.4% 0.2%Uncertainty ±0.8% ±1.1% ±0.6% ±0.5% ±1.3% ±1.2% ±0.2%

Table 8.6 Electron Neutrino Background. The νe + νe CC background per energy binas a function of the interaction type. The rows in this table represent the interaction type.

175

Chapter 9

Discussion And Final Remarks

The number of selected events (Nνµ) found in section 8.2 is a function of the neutrino flux,

and the cross section between the incident neutrinos and the number of target nuclei (T )

[113]:

Nνµ(Eν) = Tσ(Eν)Φ(Eν). (9.1)

To investigate which of the two considered quantities (σ(Eν) or Φ(Eν)) is responsible for

the excess of neutrino events predicted by the MC simulation, as discussed in section 8.2,

the following sections present the calculations of both quantities using the results already

obtained.

9.1 Flux Calculation

The flux calculation presented in this section assumes that the cross sections used by GENIE

to simulated neutrino interactions are in agreement with the current measurements1. The

flux of charged kaons produced at the NuMI target can be constrained using neutrinos

detected with NDOS since about 93% of the neutrinos in the 2 GeV peak come from charged

kaon decays. Figure 9.1 shows the longitudinal momentum (figure 9.1a) and the transverse

momentum (figure 9.1b) of the charged kaons produced at the NuMI target as a function of

1See section 9.1.2 for a discussion on the cross sections used in the MC simulation.

176

Energy (GeV)ν 0 2 4 6 8 10

(G

eV/c

)L

p±K

0

5

10

15

20

25

30

0

20

40

60

80

100

True Energy. NDOS MC.ν Longitudinal Momentum vs. ±K

(a)

Energy (GeV)ν 0 2 4 6 8 10

(G

eV/c

)T

p±K

0

0.5

1

1.5

2

2.5

3

1

10

210

310

True Energy. NDOS MC.ν Transverse Momentum vs. ±K

(b)

Figure 9.1 Charged Kaon Longitudinal And Transverse Momenta. (a) Longitudinaland (b) transverse momenta of charged kaons produced at the NuMI target as a functionof the daughter neutrino energies. Event selection criteria applied to these MC simulatedsample. MC simulation.

the daughter neutrino energy2. The average longitudinal momentum (pL) of charged kaons

when the daughter neutrino energy is restricted to:

1.5 GeV < Eν < 3.5 GeV, (9.2)

is 6.61 GeV/c. The average transverse momentum (pT ) of charged kaons for the neutrino

energies shown in equation (9.2) is 0.34 GeV. The average momentum of these charged

kaons is therefore 6.62 GeV/c. The neutrinos detected with NDOS at 110 mrad off NuMI

axis, thus, constrain the flux of charged kaons produced at the NuMI target with average

momentum of (6.62 ± 1.72) GeV/c.

The sum of all the neutrinos coming from: charged kaon, charged pion, muon, and KL

decays was presented in figure 3.10. In addition, the neutrinos coming from charged kaon

decays are dominant for energies higher than 1.5 GeV, as shown in figure 9.2. This figure is

an updated version of figure 3.12 after all the event selection is applied to the MC simulated

2Here K means K+ and K− added together.

177


Rat

ioπ

K/

0

5

10

15

20

Ratio. NDOS MC.π True Energy. K/ν

Figure 9.2 Ratio Of Neutrinos From Charged Kaon Decays Over Neutrinos FromCharged Pion Decays After Event Selection Criteria Applied. MC simulation.

sample. For these energies, the overall (integrated) background of neutrinos coming from

charged pion decays is 7.7%, and from muons +KL decays is 1.1%. Table 9.1 summarizes the

percentages of charged pions and muons + KL background per energy bin. The statistically

significant data in figure 8.16 have energies less than 3.5 GeV, so the charged kaons flux

measurement is calculated using data between 1.5 GeV and 3.5 GeV.

Decay\E (GeV) 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0

π+ → νµ, QE 62.9% 20.6% 5.7% 3.1% 2.1% 6.3% 33.3%π+ → νµ, Non-QE 37.5% 13.6% 4.1% 2.6% 3.4% 3.0% 1.1%π− → νµ, QE 40.5% 17.9% 8.9% 7.8% 19.1% 7.4% 33.3%

π− → νµ, Non-QE 39.6% 8.0% 3.0% 2.6% 4.0% 5.5% 3.2%µ− or KL → νµ, QE 2.2% 1.1% 0.6% 0.5% 1.1% 1.1% 0

µ− or KL → νµ, Non-QE 1.7% 0.8% 0.4% 0.3% 0.6% 0% 0µ+ or KL → νµ, QE 1.9% 1.2% 1.1% 0.5% 0.5% 1.0% 0

µ+ or KL → νµ, Non-QE 0.8% 0.2% 0.2% 0.5% 0.3% 0.5% 0

Table 9.1 Background To Neutrinos From Kaon Decays. Percentage of neutrinosnot coming from charged kaon decays per energy bin. The rows in this table represent thedecaying particle that produces the neutrino.

178

int.\E (GeV) 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0

νµQE 9.8 9.9 13.8 10.8 2.3 1.3 0.2νµ Non-QE 1.3 14.9 35.6 31.5 7.4 4.7 1.5νµQE 6.5 5.2 3.1 2.4 1.3 1.0 0.1

νµ Non-QE 0.9 4.8 4.5 3.6 3.0 1.8 0.6

NbinK 18.5+7.5

−3.1 34.7+7.3−5.6 57.1+10.8

−8.6 48.3 ± 9.2 14.0+4.8−2.9 8.7+5.1

−2.7 2.4+1.6−1.2

Table 9.2 Candidate Events From Kaon Decays. Number of candidate events perneutrino type and interaction type, when the parent meson is a charged kaon. Each rowin this table represents a neutrino type associated with an interaction type. The last row,Nbin

K , is the total number of selected candidate events per energy bin. These numbers comeafter NC, νe + νe, and charged pions + muons + KL background subtraction. Tables 8.5and 8.6 summarize these backgrounds.

The charged kaon decays of interest are:

K+ → νµ + µ+,

→ νµ + µ+ + π0,

K− → νµ + µ−,

→ νµ + µ− + π0. (9.3)

To find the total number of candidates coming from charged kaon decays (NK), the charged

pions + muons + KL background is subtracted from NT (see equation (8.13)). Table 9.2

summarized the various NbinK . The flux is a function of: the number of νµ + νµ, NK ,

detected with NDOS, their CC cross sections (σ) with respect to the nuclei in NDOS, the

number of target atoms (T ) in the VR, and the reconstruction efficiency (ǫ) [113]:

Φ =∑

i

NKi

ǫiσiT. (9.4)

All these quantities are a function of the energy bin, i. The total flux, Φ, is the sum of the

179

individual fluxes per energy bin.

9.1.1 Number Of Atoms In The Target Region

The number of atoms available to interact with neutrinos in the VR is calculated using

NOνA design information [114, 83], interacting with NOνA PVC extrusions experts [115],

and measuring dimensions and weights of spare NDOS PVC extrusions kept at Argonne

National Laboratory. This number of atoms is confined to the NDOS target region (TR)

with coordinates restricted by:

|X| < 106 cm,

|Y | < 172 cm,

288 cm < Z < 452 cm. (9.5)

The PVC mass inside the TR (mpvc) is 4.90% of the total PVC mass, Mpvc:

• Mpvc = (38.3 ± 1.9) × 103 Kg.

• mpvc = (1.9 ± 0.1) × 103 Kg.

The scintillator mass inside the TR (ml) is 4.94% of the total scintillator mass, Ml:

• Ml = (91.3 ± 4.6) × 103 Kg.

• ml = (4.5 ± 0.2) × 103 Kg.

Even though the amount of WLS fiber, glue, and air within the TR is very small, it is

calculated as well:

• WLS fiber mass: mwls = (5.1 ± 0.3) Kg.

180

• Glue mass: mg = (3.1 ± 0.2) Kg.

• Air mass: ma = (6.0 ± 0.3) × 10−3 Kg.

The NOνA software has a function, GeometryBase::TotalMass, which provides the numbers

discussed above:

• MC simulated PVC mass: MMCpvc = 39.9 × 103 Kg.

• MC simulated scintillator mass: MMCl = 87.7 × 103 Kg.

Component Molar Mass (g) fraction (f) Mass (Kg) Moles Molecules

Scintillator

CH2 14 0.9464 4270.6 3.02×105 1.83×1029

C9H12 120 0.0523 236 1.97×103 1.18×1027

C15H11NO 221 0.0014 6.3 2.84×10 1.71×1025

C24H22 310 0.0002 0.7 2.33 1.4×1024

PVC

C2H3Cl 62 0.8 1502 2.42×104 1.46×1028

TiO2 80 0.15 281.6 3.52×103 2.12×1027

C36H70O4Ca 607 0.006 11.3 1.86×10 1.12×1025

C20H24 264 0.009 16.9 6.4×10 3.85×1025

C12H20O5 244 0.002 3.8 1.54×10 9.27×1024

C21H42O4 359 0.002 3.8 1.05×10 6.3×1024

C5H8O2 100 0.031 58.2 5.82×102 3.5×1026

Air

N 14 0.78 5.0×10−3 0.33 2.01×1023

O 16 0.21 1.0×10−3 0.08 4.72×1022

Ar 40 0.01 1.0×10−4 1.35×10−3 8.13×1020

WLS Fiber

C6H5CH 90 1 5.1 5.68×10 3.42×1025

Glue

C 12 1 3.1 2.59×102 1.56×1026

Table 9.3 Chemical Composition Of The Prototype Detector. NDOS chemical com-position, and number of atoms or molecules in each of the elements or compounds that areinside the target region defined in equation (9.5).

The chemical composition of the NDOS components [116] is summarized in table 9.3.

With the identity of all the molecules present in the NDOS compounds, the procedure to

181

obtain the target mass is straight forward. The molar mass (mm) of molecules or atoms

is the mass of one mole of such substance. Table 9.3 contains the proportions (f) of each

element or molecule in each of the compounds of NDOS. The number of target atoms or

molecules (Tx), inside the TR, of a given element or compound with mass mx is:

Tx =mxfna

mm, (9.6)

where na is Avogadro’s number. The known cross sections are for individual atoms, hence

the number of atoms in each molecule is known from its chemical composition. Table 9.3

summarizes the number of atoms or molecules that make up the mass of the NDOS TR.

9.1.2 Monte Carlo Cross Sections

GENIE has tables with the cross sections of many processes labeled under the categories:

QE and non-QE, where GENIE labels the individual non-QE processes as: resonance, deep

inelastic or coherent. Such tables contain the various cross sections as a function of the

neutrino energy as well as the target nucleus. Since the neutrino energies recorded in the

tables are discrete, GENIE interpolates the recorded cross sections in order to obtain the

cross section of a particular event for the given energy. The GENIE cross sections used in

this analysis are displayed in figure 9.3. All the non-QE cross sections per nucleus are added

together.

The cross sections shown in figure 9.3 are calculated assuming that experimental results

on cross sections per nucleon can be extrapolated to give nuclei cross sections. The uncer-

tainties given by GENIE to these cross sections are around 10%, from the green band shown

in figure 9.4. This figure also summarizes the measurements used to calculate the GENIE

182

(a) (b)

Figure 9.3 GENIE Cross Sections As A Function Of Neutrino Energy. Cross sectionfor each of the elements present in the NDOS target region’s mass. The color code is in theright side of the figures, solid lines are for neutrinos and dashed lines are for antineutrinos.Summary of (a) QE and (b) non-QE cross sections.

Figure 9.4 GENIE Uncertainty In The Muon Neutrino Cross Section. νµ CCscattering from an isoscalar target. The shaded band indicates the estimated uncertainty onthe free nucleon cross section [109].

183

cross sections: [117], [118], [119], [120, 121], [122, 123], [124], [55], and [125].

9.1.3 Total Reconstruction Efficiency

The last piece to find the sought flux is the reconstruction efficiency. This efficiency was

calculated using the MC simulated sample, and reflects the fraction of events reconstructed

given the total number of events available:

ǫ =NR

NMC, (9.7)

where NR is the number of reconstructed MC simulated events that passed all the event

selection cuts, and NMC is the number of MC simulated events which had their interaction

point inside the VR. The reconstruction efficiency as a function of the neutrino energy, and

discriminated per neutrino type and interaction type is presented in figure 9.5.

Energy (GeV)ν 1.5 2 2.5 3 3.5

Effi

cien

cies

0

0.1

0.2

0.3

0.4

0.5

0.6Reconstruction Efficiencies. NDOS MC.

QEµν Non-QEµν QEµν Non-QEµν

Figure 9.5 Reconstruction Efficiency As A Function Of The Neutrino Energy. Errorbars represent the binomial uncertainties. MC simulation.

184

9.1.4 νµ + νµ Flux Coming From Charged Kaon Decays

Energy (GeV)1.5 2 2.5 3 3.5

)2 /

PO

T c

m-1

1F

lux

(10

-110

1

10

Data

MC

: Flux vs. Energy. NDOS MC & DATA.ν

Figure 9.6 Total Flux Of νµ + νµ Coming From Charged Kaon Decays. NDOS data(black) and MC simulation (red).

All components of equation (9.4) have been presented so that the flux of νµ+ νµ produced

in charged kaon decays can be determined. The cross sections are given discriminated by

interaction type and neutrino type, therefore there are four fluxes:

Φαβi =

Nαβi

ǫαβi σ

αβi T

; α = QE, non-QE; β = νµ, νµ, (9.8)

and the total flux is the sum of the four:

Φ =∑

α,β

∑

i

Φαβi , (9.9)

where i labels the energy bin. Here, N is the number of neutrino candidate events coming

from charged kaon decays (NK) presented in table 9.2. T is the numbers of target nuclei

185

presented in table 9.3. σ are the cross sections presented in figure 9.3, and ǫ are the recon-

struction efficiencies presented in figure 9.5. The total flux of νµ + νµ coming from charged

kaon decays is shown in figure 9.6 in four energy bins of 0.5 GeV size. The uncertainties are

the sum in quadrature of statistical and systematic uncertainties. The systematic uncertain-

ties are summarized in figure 9.7. The flux calculated from the NDOS data is systematically

lower than that predicted by the NDOS MC simulation (see figure 8.16), within the energy

region of interest defined in equation (9.2). The χ2 test for these distributions concludes

that χ2 = 9.336 for ndf = 4. The ratios between data and MC simulation for the total

flux, and also for QE and non-QE interactions, are shown in figure 9.8. The average ratio is

about 0.7.

Energy (GeV)ν1.5 2 2.5 3 3.5

Unc

erta

inty

(%

)

-10

-5

0

5

10

15

20Flux Systematic Uncertainties. NDOS MC.

Num. of eventsCross Sec.Reco. Eff.POTNum. Of Targets

Figure 9.7 Flux Systematic Uncertainties. MC simulation.

Figure 9.9 presents the fluxes discriminated by interaction type. The agreement within

error bars of the non-QE fluxes in figure 9.9b is better than the one in figure 9.9a with the

186

Energy (GeV)ν 1.5 2 2.5 3 3.5

Rat

io

0

0.2

0.4

0.6

0.8

1 Flux Ratio: DATA / MC. NDOS MC & DATA.ν

All

QE

Non-QE

Figure 9.8 Data Over MC Simulated Flux Ratio. Total flux ratio (black), QE flux ratio(blue), and non-QE flux ratio (red) as a function of neutrino energy.

Energy (GeV)0.5 1 1.5 2 2.5 3 3.5

)2 /

PO

T c

m-1

1F

lux

(10

-210

-110

1

10

Data

MC

: Flux vs. Energy. NDOS MC & DATA. QE.ν

(a)

Energy (GeV)0.5 1 1.5 2 2.5 3 3.5

)2 /

PO

T c

m-1

1F

lux

(10

-110

1

10

Data

MC

: Flux vs. Energy. NDOS MC & DATA. Non - QE.ν

(b)

Figure 9.9 Total Flux Of νµ + νµ Coming From Charged Kaon Decays Discrim-inated By Interaction Type. (a) QE and (b) non-QE fluxes. NDOS data (black) andMC simulation (red).

187

QE fluxes, though the systematically lower measured flux is present in both interactions.

This is corroborated by comparing the two χ2 tests:

χ2QE = 6.506, χ2

NQE = 3.995, ndf = 4,

χ2NQE

χ2QE

= 0.614. (9.10)

Equation (9.10) shows a closer agreement between the non-QE data and the prediction by

the MC simulation than the QE data does. This result was expected since the non-QE data

sample is 60% larger than the QE one, which is a big difference in statistics, a determinant

factor in the analysis. The discrimination between QE and non-QE flux is useful to check if

the two results are consistent with each other, i.e. the flux must be the same, regardless of

the interaction, since the difference between them comes from the cross sections. The ratio

between the QE over the non-QE fluxes in figure 9.10, is consistent with 1, as expected.

Energy (GeV)ν0.5 1 1.5 2 2.5 3 3.5

QE

/ N

on-Q

E

0

0.5

1

1.5

2Flux Ratio: QE / Non - QE. NDOS DATA.

Figure 9.10 QE Over Non-QE Flux Ratio.

188

The flux of charged kaons produced at the NuMI target can be constrained comparing

the data presented above. From the neutrinos detected with energies stated in equation

(9.2), the flux of charged kaons with total average momentum of 6.62 GeV/c compares to

that predicted by the MC simulation as follows:

ΦData = (1.462 ± 0.154 stat+0.157−0.113 syst) × 10−10 ν

cm2POT,

ΦMC = (2.086+0.141−0.139) × 10−10 ν

cm2POT,

ΦData = 0.701+0.108−0.094 · ΦMC (9.11)

The observed data and the MC simulated distributions shapes agree, as seen in figures 9.6

and 9.9, though their normalization is not the same, these differ by the amount presented

in equation (9.11). This analysis suggests that the neutrino flux of νµ from charged kaon

decays obtained from the NDOS data is 30% lower that is predicted by the MC simulation.

9.2 Inclusive νµ Charged Current Cross Section Calcu-

lation

With the results presented in section 9.1, it becomes interesting to calculate the inclusive

νµ CC cross section using the selected candidates3 (Nνµ) presented in table 8.3, and the

flux (Φ) predicted by the MC simulation. With the direct measurement of the number of

events done in this analysis it is possible to calculate either the flux, or the cross section,

using previous measurements incorporated in the MC simulation of the cross section or

the flux, respectively. The excess of predicted events observed in the MC simulation (see

3The same procedure that led to equation (8.13) applies here for the number of νµ, Nνµ .

189

section 8.2) could be the result of overestimating either the flux or the cross section, or a

more complicated combination of over and underestimation of the two. A manipulation of

equation (9.4) results in the following relation between the cross section and the flux [113]:

σ =∑

i

N iνµ

ǫiΦiT. (9.12)

The two previous direct measurements (see section 2.1.1.4) of the inclusive νµ CC cross

section around 2 GeV (refer to figure 2.4) reported their results per nucleon, therefore the

number of targets (T ) used in section 9.1.1 needs to be slightly modified from the number

of target nuclei to the number of target nucleons. The inclusive cross section per nucleon

(σN ) is the sum of the QE cross section per nucleon (σQE) plus the non-QE cross section

per nucleon (σnQE):

σN = σQE + σnQE. (9.13)

The number of targets in the TR is:

T = 1.723 × 1030neutrons, or

T = 3.854 × 1030nucleons. (9.14)

The FLUKA νµ flux prediction (Φi), that is embedded in the MC simulated sample, is

shown as a function of neutrino energy in figure 9.11. The same FLUKA simulation was

used by the MiniBoone collaboration on their measurement of νµ and νe events [126]. The

MiniBoone detector is also located at 110 mrad off the NuMI beam axis. Their measure-

ments have a 9% systematic uncertainty, on the NuMI flux, in the energy range between

0.9 GeV and 3.0 GeV. In this analysis the systematic uncertainty related to the FLUKA flux

190

Energy (GeV)ν 1.6 1.8 2 2.2 2.4

)2 /

PO

T c

m-1

1F

lux

(10

0

1

2

3

4

5

6

QE

Non-QE

Flux. NDOS MC. QE & Non-QE.µν

Figure 9.11 Muon Neutrino MC Simulated Flux Prediction.

Energy (GeV)ν 1.6 1.8 2 2.2 2.4

Effi

cien

cies

0

0.1

0.2

0.3

0.4

QE

Non-QE

Reconstruction Efficiencies. NDOS MC.

Figure 9.12 Reconstruction Efficiency As A Function Of The Neutrino Energy.Error bars represent the binomial uncertainties. MC simulation.

191

prediction is assumed to be 10% based on the MiniBoone estimation.

Energy (GeV)ν1.6 1.8 2 2.2 2.4

/ nu

cleo

n)2

cm

-38

Cro

ss S

ectio

n (1

0

0

0.5

1

1.5

2

2.5

DataMC

Energy.ν Charged Current Cross Section vs. µνInclusive

Figure 9.13 Inclusive Muon Neutrino Charged Current Cross Section Per NucleonOn A Carbon Target. NDOS data (black) and MC simulation (red).

The last pieces to calculate the inclusive νµ CC cross section are the reconstruction

efficiencies (defined in equation (9.7)) presented in figure 9.12. Following a procedure similar

to that shown in section 9.1.4 leads to the inclusive νµ CC cross section per nucleon on

a carbon target4. The cross section is presented in two energy bins of 0.5 GeV, as seen

in figure 9.13. The uncertainties are the sum in quadrature of statistical and systematic

uncertainties. The systematic uncertainties are summarized in figure 9.14. The cross section

calculated using the NDOS data is systematically lower than that embedded in the NDOS

MC simulation. The χ2 test for these distributions concludes that χ2 = 6.803 for ndf = 2.

The ratio between the data and MC simulated cross sections is shown in figure 9.15, only

the error bars in the second bin are consistent with one, which is shown as a dashed line.

4Refer to table 9.3.

192

Energy (GeV)ν1.6 1.8 2 2.2 2.4

Unc

erta

inty

(%

)

-20

-10

0

10

20Cross Section Systematic Uncertainties. NDOS MC.

Num. of EventsReco. Eff.FluxNum. Of TargetsPOT

Figure 9.14 Cross Section Systematic Uncertainties. MC simulation.

Energy (GeV)ν1.6 1.8 2 2.2 2.4

Rat

io

0.4

0.6

0.8

1

1.2

CC Cross Section Ratio: Data / MC. NDOS MC & DATA.µνInclusive

Figure 9.15 Data Over MC Simulated Muon Neutrino Charged Current CrossSection Ratio.

193

Figure 9.16 Measurements Of Muon Neutrino Charged Current Inclusive Scat-tering Cross Sections Divided By Neutrino Energy. All the results are cited in thefigure. Vertical blue (red) error bars represent statistical (combined statistical and system-atic) errors. The NDOS data point is inserted into a figure taken from [44].

The inclusive νµ CC cross section per nucleon at an average neutrino energy of 1.97 GeV

compares to that embedded in the MC simulation as follows:

σData = (1.085 ± 0.219 stat+0.330−0.383 syst) × 10−38cm2,

σMC = (1.588+0.216−0.210) × 10−38cm2,

σData = 0.683+0.266−0.257 · σMC. (9.15)

The comparison of the measured cross section with previous measurements is presented in

figure 9.16, which summarizes the measurements of νµ CC inclusive scattering cross sections

divided by the neutrino energy as a function of neutrino energy. Note the transition between

logarithmic and linear scales occurring at 100 GeV. The red error bars are the sum in

quadrature of the statistical and systematic uncertainties. The blue error bars are the

194

statistical uncertainties alone. The measured cross section agrees, within error bars, with the

two previous results [55, 56] mentioned above, and lies below the trend set by the collection

of all the results shown. The average cross section over neutrino energy of the wide range of

energies shown is represented by the horizontal dashed line.

9.3 Final Remarks

The two results presented in equations (9.11, 9.15) are the possible consequences of the

difference of about 1/3 seen in section 8.2 between the selected NDOS νµ+ νµ CC candidates

and the prediction done by MC simulation. The conclusion is that the FLUKA flux needs to

be adjusted, based on the two NDOS measurements available5, by 30% since the inclusive

νµ CC cross section (over neutrino energy) embedded in the MC simulation is consistent

with the dashed line in figure 9.16 at 2 GeV.

NDOS was designed to serve as a testing instrument that would provide the NOνA

collaboration with a proof of principle for the concepts and technologies developed for the

experiment. Measuring the flux of νµ + νµ coming from charged kaon decays was a worth

pursuing result since those neutrinos show a clean peak around 2 GeV, and the flux of

charged kaons had not been measured at the angle where NDOS sits. A 30% excess of

events predicted by the MC simulation is now reported by two different analyses of the

NDOS data, which should be taken into account by the neutrino oscillations community.

NOνA can verify the results presented in this analysis with its near detector underground

which is taking data now.

5The result of this analysis and that from [22].

195

APPENDICES

196

Appendix A

Mass Terms In The Weak Interaction

All leptons participate in the weak interaction. However it is the only interaction in the

Standard Model where neutrinos are involved1. Leptons and quarks enter in the weak

interaction as left-handed weak-isospin doublets2:

L ≡(

ν

l

)

, q ≡(

u

d

)

, (A.1)

where the left-handed states are:

νL =1

2(1 − γ5)ν, lL =

1

2(1 − γ5)l,

uL =1

2(1 − γ5)u, dL =

1

2(1 − γ5)d. (A.2)

Here: u = u, c, t; d = d, s, b; l = e, µ, τ ; and ν = νe, νµ, ντ . There is no experimental

evidence of the existence of right-handed neutrinos3, and since these only interact with other

1Neutrinos do not participate in the electromagnetic interaction at the tree level. Ifneutrinos are Dirac particles these can have a magnetic moment, and therefore participatein the electromagnetic interaction through one-loop diagrams. The transition moment, whichis relevant to νi → νj + γ (i 6= j), may exist for both Dirac and Majorana neutrinos [64].For a detailed calculation of the one-loop diagram see [127, 128, 129].

2The doublet in equation (A.1) is a general case which works for all three families.3Right-handed neutrinos have chirality eigenvalue: +1. Chirality is an intrinsic property,

independent of the reference frame of the observer. Since neutrinos are massive, they canhave both eigenvalues of helicity: ±1, which is a property that depends on the referenceframe of the observer. Chirality and helicity are only equivalent for massless particles [130].

197

particles through the weak interaction4, the Standard Model excludes them. Right-handed

charged leptons and right-handed quarks enter the Standard Model weak interaction as

weak-isospin singlets given by:

lR =1

2(1 + γ5)l,

uR =1

2(1 + γ5)u, dR =

1

2(1 + γ5)d. (A.3)

Weak-isospin and weak hypercharge are related by the Gell-Mann–Nishijima [131, 132] rela-

tion:

Q = I3 +Y

2, (A.4)

where Q is the electric charge, I3 is the third component of the weak-isospin, and Y is the

weak hypercharge. Weak-isospin can only take two values I3 = ± 1/2, thus:

YL = −1, YR = −2. (A.5)

With the commutator:

[I3, Y ] = 0, (A.6)

the two quantities are commuting observables, and the product of the group transformations

generated by I and Y is the gauge group SU(2)L ⊗ U(1)Y of a gauge theory. Four massless

bosons and six massless leptons are the result of these constructions [130]. The Lagrangian

of this group is:

LG = Lg + Lφ + Lf + Lh, (A.7)

4The weak interaction only operates on left-handed states [27].

198

where Lg is the gauge Lagrangian, Lφ is the scalar Lagrangian, Lf is the fermionic La-

grangian, and Lh is the Yukaga Lagrangian. The gauge part of the Lagrangian is:

Lg = −1

4W i

µνWµνi − 1

4BµνB

µν , (A.8)

where W iµν is the SU(2) gauge field that has three and four-point self-interactions, and Bµν

is the U(1) gauge field associated with Y , which has no self-interaction [130].

The scalar part of the Lagrangian is:

Lφ = (Dµφ)†Dµφ− V (φ), (A.9)

where Dµ is a covariant derivative [133], and V (φ) is a scalar potential. In order to generate

the masses of the weak bosons and all the charged leptons, a doublet of scalar fields (φ)

defined as:

φ ≡(

φ+

φ0

)

, (A.10)

is introduced. The known masses are generated through a spontaneous symmetry breaking

in the potential V (φ) of the field. This transforms as a SU(2)L doublet which implies:

Yφ = 1. The invariance and renormalizability [134] of the group restrict V (φ) to the form:

V (φ) = µ2φ†φ+ |λ|(φ†φ)2, (A.11)

where µ and λ are parameters. For µ > 0 the potential is quadratic with its only minimum at

zero, as shown in figure A.1a, and the vacuum is empty. An exact symmetry is characterized

by two conditions: the Lagrangian density is invariant under the symmetry, and the physical

199

vacuum is invariant under the symmetry transformations [130]. A model with an exact

symmetry contains a degenerate set of massless particles. The Standard Model has symmetry

breaking to account for the observed masses in the model. The case for µ < 0, in equation

(A.11), is shown in figure A.1b. There is a spontaneous symmetry breaking since the scalar

field takes one of all the possible minima in the potential.

(a) (b)

Figure A.1 Scalar Potential. (a) µ > 0, and (b) µ < 0 [135].

The fermionic part of the Lagrangian is:

Lf =3∑

k=1

(

qkLı6DqkL + lkLı6DlkL + ukRı6Duk

L + dkRı6DqkR + lkRı6DlkR

)

, (A.12)

where k denotes the family, and all fields are weak eigenstates [130].

200

The Yukawa part of the Lagrangian is:

Lh = −3∑

k,n=1

[

Γuknu

nR

(

ıτ2φ†qkL)

+ Γdknd

nR

(

ıτ2φ†qkL)

+ Γlknl

nR

(

ıτ2φ†lkL)]

+ h.c., (A.13)

where the matrices Γkn describe the Yukawa couplings between the scalar doublet and the

various flavors k and n of quarks and leptons [130].

The vacuum expectation values of the various complex scalar fields are the components

φi of a complex vector υ:

υ = 〈0|φ|0〉. (A.14)

Without any loss of generality the four axes of this space can be chosen so that:

〈0|φi|0〉 = 0, for i = 1,2, and 4, and 〈0|φ3|0〉 = v. (A.15)

Here υ is determined by writing the scalar potential: V (φ) → V (υ), and choosing υ such

that V is minimized [136]. The quantum theory is obtained by considering the fluctuations

around this minimum: φ = υ + φ′. Using equation (A.15), the potential in equation (A.11)

becomes:

V (φ) → V (υ) =1

2µ2v2 +

1

4|λ|v4. (A.16)

For µ > 0 in figure A.1a, the minimum of the potential occurs at v = 0, and no symmetry

breaking occurs. For µ < 0 in figure A.1b, the minimum is obtained at a non-zero value of

v:

dV (v)

dv= v(µ2 + |λ|v2) = 0, (A.17)

201

which has solution5:

v =

(

−µ2

|λ|

)1/2

. (A.18)

With this result, the kinetic part of the scalar Lagrangian in equation (A.9) becomes:

(Dµφ)†Dµφ =1

2(0 v)

[

g

2τ iW i

µ +g′

2Bµ

]2(0

v

)

+ · · · , (A.19)

where g and g′ are coupling constants of the SU(2) and U(1) gauge groups. After some

algebraic manipulation [138], important mass terms are identified6:

1 :g2v2

8(W 1 − ıW 2)µ(W 1 + ıW 2)µ, (A.20)

2 :(g2 + g′2)v2

8

(

gW 3

√

g2 + g′2− g′B√

g2 + g′2

)µ(gW 3

√

g2 + g′2W 3 − g′B

√

g2 + g′2

)

µ

.

The masses of the two weak interaction bosons are:

MW =gv

2, MZ =

v√

g2 + g′2

2, (A.21)

and the weak mixing angle (θW ) fulfills:

cos θW =g

√

g2 + g′2, sin θW =

g′√

g2 + g′2. (A.22)

1 and 2 in equation (A.20) can be written as:

1 : M2WW+µW−

µ , 2 :M2

Z

2ZµZµ, (A.23)

5The case for µ = 0 requires loop corrections, but the symmetry is again spontaneouslybroken [137].

6These are mass terms since the fields couple to themselves.

202

where:

W± =1√2(W 1 ∓ ıW 2), and Z = cos θWW 3 − sin θWB, (A.24)

are the weak interaction bosons. From the form of the covariant derivative, the electron’s

charge is [133]:

e =gg′

√

g2 + g′2, (A.25)

which allows to estimate the weak interaction bosons’ masses. Notice that the term:

AµAµ = (sin θWW 3 + cos θWB)µ(sin θWW 3 + cos θWB)µ, (A.26)

is multiplied by the zeroes in equation (A.19), thus the photon field (A), given by:

A = sin θWW 3 + cos θWB, (A.27)

is massless. The coefficients MW and MZ (in equation (A.23)) depend on v, which is7 the

vacuum expectation value of the scalar field. This scalar field is known as the Higgs field

[139]. The spontaneous symmetry breaking that results in v 6= 0 produces three massive

weak interaction messengers and a massless electromagnetic interaction one [130]; unifying

the weak and electromagnetic interaction into one elegant electroweak interaction.

7See equations (A.15, A.18).

203

Appendix B

NOνA Kalman Tracker

For track propagation in the Kalman Tracker, each track is represented as a system with

two parameters: position and slope. A minimum of two cell hits are required to estimate the

parameters. Track seeds are formed from pairs of cell hits separated by as much as 3 planes

in their view. The track starting point is assumed to be at the center of the cell hit with the

highest Z coordinate, and the track propagation is carried on towards lower Z coordinates.

Tracks are propagated plane by plane, using the current estimates of the track position and

slope, to estimate the location of the expected track’s cell hits in the next plane. For cell

hits on the projected plane, a ∆χ2 test is calculated from the inclusion of the new cell hit

in the track. If the ∆χ2 is less than a default value, the cell hit is added to the track. After

a new cell hit is added to the track, a new track fit is done to replace the older position and

slope parameters. When the estimated track position for the next cell hit falls in an inactive

cell, and no other cell hits in that plane are added to the track, the plane will not count for

the propagation algorithm. The track propagation continues until three consecutive planes

with no cells hits are found.

When multiple cell hits in a given plane are considered to be part of a track, the addition

of each cell hit is determined individually and independently from the inclusion of the other

cell hits on the plane. When multiple cell hits are added to a track on one plane, the

propagation to the next plane is calculated based on the average of all estimated states

204

found from adding each cell hit individually. After each propagation step the resulting

tracks are filtered. At the initial propagation, the filter provides the best estimated of the

track state for the propagation in the opposite direction. At the final propagation, the filter

provides the best estimate of the fit to the track.

Figure B.1 Sample Event. Illustrates the track finding process. MC simulation.

Before a 2D track is written in the output the following quality cuts are applied:

• Minimum number of cell hits per track: 4.

• Minimum number of valid planes crossed by the track: 3.

• Maximum χ2 per added cell hit: 10.

The cuts reject fake tracks or tracks with poor fits. From the tracks that pass the quality

control, the best track found is the one with the lowest χ2/Nch, where Nch is the number

of cell hits. As mentioned in chapter 4, long muon tracks are expected to undergo multiple

scattering at the end of their path. This feature will raise the χ2/Nch, which is taken into

205

account. The best 2D track found is written to the output, and its cell hits are removed

from the pool of cell hits in that particular slice. The entire 2D track finding is repeated

with the remaining cell hits in the slice to find new tracks.

The algorithm stops when no new tracks pass the cuts, or there are no more available

cell hits in the slice. A sample output of the algorithm is presented in figure B.1. From left

to right: the first iteration finds one track per view (red and green), the second iteration

finds one track per view (green and yellow), and the third iteration finds only one track in

the side view (light blue). A flow chart of the algorithm is shown in figure B.2.

Figure B.2 Flow Chart. Shows how the algorithm finds 2D tracks [140].

206

Within a single slice, 2D tracks from separate views are matched together to form 3D

tracks based on a scoring routine. From each pair of 2D tracks from separate views that

overlap in their Z coordinates, a match score is calculated based on the starting and ending

Z coordinates of each track:

score =|Zlt − Zls| + |Zht − Zhs|

OZ. (B.1)

Here Zlt is the lowest Z coordinated of the 2D track in the top view, Zls is the lowest Z

coordinated of the 2D track in the side view, Zht is the highest Z coordinated of the 2D

track in the top view, Zhs is the highest Z coordinated of the 2D track in the side view, and

OZ is the length of the overlap of the two tracks in the Z coordinate. The two 2D tracks

with the lowest score are matched in each iteration to form a 3D track. The algorithm is

repeated until no more 2D tracks overlap in the Z coordinate. Each 2D track is only allowed

to be matched with one 2D track on the other view. All 3D tracks and unmatched 2D tracks

are written to the output. The final output of tracks from the sample event in figure B.1 is

presented in figure B.3. Two 3D tracks are reconstructed (red and green), and one 2D track

is left unmatched (blue). For a full description of the Kalman Tracker see [140].

207

Figure B.3 Sample Event. Final output of tracks from the sample event in figure B.1.

208

Appendix C

Unfolding Algorithm, TSVDUnfold

One of the many uses of the MC simulated sample is that the reconstructed neutrino en-

ergies can be compared to the corresponding true energies. Figure 6.14, for example, is a

comparison of true and reconstructed neutrino energies. From this figure it is clear that

some events do not have their reconstructed energy in the appropriate energy bin. There

are various reason for this to happen, most of them already mentioned. For example figures

6.10a and 6.10b show that muons with true energy less than 0.4 GeV attain a reconstructed

energy 25% higher than the true energy. A solution to this issue comes from the use of an

unfolding algorithm [141]. This is based on the knowledge of both true and reconstructed

energies, which allows to take into account the inefficiencies and resolutions of the detector

to try to place the reconstructed events in the appropriate energy bin.

Singular Value Decomposition Unfolding Algorithm

The unfolding algorithm incorporates the Singular Value Decomposition (SVD) method to

find the detector’s response matrix. The inputs to the algorithm are [141]:

• ~x ini: Vector of simulated events.

• ~b ini: Vector of reconstructed simulated events.

209

The components of the ~x ini vector are the MC simulated events that realistically represent

the underlying physics of the studied situation. The components of the ~b ini vector are

reconstructed MC simulated events which contain information about the performance of the

detector. The goal of the algorithm is to find a probability matrix A that will relate the two

vectors [141]:

A~x ini = ~b ini. (C.1)

Once the matrix is know, the algorithm performs the unfolding procedure that takes the real

life measurements gathered in the vector ~b to an unfolded vector ~x that will represent the

most probable distribution of events given the performance of the detector [141]:

A~x = ~b. (C.2)

To obtain the vector ~x requires the knowledge of the determinant of A, which might not be

a straightforward endeavor since the components of the matrix could have values ranging

from big numbers to very small numbers. The inverses of these numbers could introduce a

wide variety of rapidly oscillating solutions to the mathematical problem. The SVD method

is a powerful tool that provides useful information about the matrix, e.g. if A is orthogonal,

all its singular values are 1. In this method the matrix A is represented as the product of

three matrices1 [141]:

A = USV T , (C.3)

where U and V are orthogonal matrices, and S is a diagonal matrix with non-negative

entries called singular values. These represent the reconstruction efficiencies per energy bin.

1V V T = 1 = V TV , UUT = 1 = UTU .

210

As mentioned above, if some of these singular values are small, the system of equations could

become very difficult to solve.

With ~b and A known, solving for ~x requires the use of the SVD method [141]:

A~x = ~b = USV T~x = US~z,

UTUS~z = UT~b = ~d = S~z,

S−1~d = ~z = V T~x,

~x = V ~z = V S−1~d = V S−1UT~b = A−1~b. (C.4)

The solution means that ~b is a set of orthonormalized functions of a parameter i = 1, ..., nb.

The basis of these functions are the columns of the matrix U , and the elements of ~d are the

coefficients of the decomposition. The vector ~x is decomposed in a series of orthonormalized

functions of a parameter j = 1, ..., nx. These functions gather in the columns of the matrix

V . The coefficients of the decomposition are the components of the vector ~z. At the end,

the initial problem is reduced to solving a diagonal system:

~z = S−1~d, (C.5)

which is just a matter of inverting the diagonal matrix S, i.e. inverting the singular values.

If some of the singular values are small, and the measurements in ~b have associated

errors, the exact solution does not give any useful information. The following 2 by 2 system

211

is presented as an example:

~b =

b1

b2

, U = V =

1√2

1 1

1 −1

, S =

1 0

0 ǫ

,

~x = V S−1UT~b =1√2

1 1

1 −1

1 0

0 1ǫ

1 1

1 −1

1√2~b,

~x = V S−1UT~b =1

2

1 1

1 −1

1 1

1ǫ

−1ǫ

~b =1

2

1 + 1ǫ 1 − 1

ǫ

1 − 1ǫ 1 + 1

ǫ

~b,

~x =1

2

(1 + 1ǫ )b1 + (1 − 1

ǫ )b2

(1 − 1ǫ )b1 + (1 + 1

ǫ )b2

=

b1 + b22

1

1

+b1 − b2

2ǫ

1

−1

,

~x = z1~V1 + z2~V2 =d1

s1~V1 +

d2

s2~V2. (C.6)

To determine if the components of the vectors ~V1 and ~V2 are statistically significant to the

problem, the following relation between the coefficients d1 and d2 needs to be true [141]:

d22 ≤ d1, (C.7)

If relation C.7 is true, the components of ~V2 are not statistically significant and d2 is simply

a random number. However, if:

s2 <1√d2, (C.8)

~V2 dominates over ~V1, and the result is meaningless. Since the si are the efficiencies per

energy bin, what this means is that with very low efficiencies, the correct bin for a particular

212

event can not be resolved between bin 1 or 2.

In the general case, for a smooth distribution of measurements, only the first few di are

statistically significant. In a plot of log|di| vs. i, the values of di should exponentially decrease

with increasing i until these reach a point of rapid oscillation, therefore the di are a series of

random numbers. Only the di occurring before the region of rapid oscillation are statistically

significant, and useful in the solution of the problem. Once the number of statistically

significant di is established, the system is regularized to avoid possible meaningless results,

as mentioned above [141].

To measure the deviations of ~x from ~xini, a vector ~ω, with components: ωj = xj/xinij , is

introduced. The product of the ωj with the columns of the probability matrix Aij results

in the number of events that were generated in bin j but belong to bin i. The result is the

matrix A, a number-of-events matrix. Rescaling the equations does not change the exact

solution. This unfolding problem is an over determined problem and its solution is obtained

by means of the least squares approach [141]:

nb∑

i=1

nx∑

j=1

Aijxj − bi

2

= min, (C.9)

whenever the errors in the measurements are all equal. Since that is not the case here, i.e.

the errors vary from bin to bin, each equation should be weighted by its measured error

[141]:nb∑

i=1

nx∑

j=1

Aijxj − bi∆bi

2

= min. (C.10)

The general equation is:(

A~x−~b)T

B−1(

A~x−~b)

= min, (C.11)

213

where B is the covariance matrix of ~b, which gathers the errors in the measurements. B is

a symmetric and positive-definite matrix, therefore its SVD is:

B = QRQT , (C.12)

with:

Rii ≡ r2i > 0 and Rij = 0, for i 6= j. (C.13)

If the statistics of the MC simulation are one or two orders of magnitude larger than the data

statistics, the unfolding error is dominated by the measurement errors, which is the case in

the this analysis. Dividing each equation by its measured error gives them equal footing in

the problem, and equation (C.11) transforms to [141]:

(

A~ω −~b)T

QR−1QT(

A~ω −~b)

= min, (C.14)

with:

Aij =1

ri

∑

m

QimAmj , bi =1

ri

∑

m

Qimbm, (C.15)

such that:

(

A~ω −~b)T

QR−1QT(

A~ω −~b)

= min =(

A~ω − b)T (

A~ω − b)

. (C.16)

Thus,

∑

i

Aijωj = bi. (C.17)

214

The regularization is achieved by introducing a stabilization term (see [142], [143], and

[144]) into equation (C.11):

(

A~ω − b)T (

A~ω − b)

+ τ · (C~ω)T (C~ω) = min, (C.18)

where τ is a stabilization parameter, and C is the a priori condition matrix. These are

both problem dependent, and control the equations that present the irregular behavior.

The components Cij are chosen such that the components of ~ω with irregular behavior are

suppressed.

The solution to equation (C.18), for any τ , requires the damped least squares approach

[144]:

AC−1

√ω I

C~ω =

b

0

, (C.19)

from which a similar result as in equation (C.5) is achieved2:

d = S−1S~d, z = S−1~d, ~ω = C−1V z, xi = xinii ωi, (C.20)

where:

si =s2i − τ

si, (C.21)

are the new singular values. The choice of τ comes from the log|di| vs. i plot. The singular

value sk of the last statistically significant equation, k, determines it:

τ = s2k. (C.22)

2Detailed calculation in [144], Chapter 25, Section 4.

215

With all the pieces of the algorithm together, the unfolding procedure is the following:

1. Define the number of bins nb, and their boundaries.

2. Define the number of bins nx, and their boundaries.

3. Build the condition matrix C introduced in equation (C.18), and calculate its inverse

C−1.

4. Generate the simulated events to find: ~x ini, reconstruct the simulated events to find:

~b ini, and construct A, which is the number-of-events matrix introduced in equation

(C.14).

5. Calculate the covariance matrix B introduced in equation (C.11).

6. Calculate Aij , and bi from equation (C.15).

7. Multiply AC−1 to fulfill equation (C.19).

8. Plot log|di| vs. i to determine the last statistically significant equation k, and set τ

(see equation (C.22)).

9. Obtain the unfolded distribution of events x from equation (C.20).

ROOT (see [145, 146]) has a tool, TSVDUnfold, that implements the unfolding algorithm

in a very user friendly way. This tool requires the user to provide:

• ~x ini.

• ~b ini.

• A.

216

With these inputs, TSVDUnfold goes through steps 1 to 8 in the algorithm. Then it asks

the user to use the log|di| vs. i plot to provide the parameter k. Based on simulated events

and reconstructed simulated events, TSVDUnfold calculates the matrices B and C. At this

point it trained itself to be able to handle real data. Now the user provides a new vector:

~bdata, which contains all the real data events. From ~bdata and a set3 of toy MC simulated

events, generated by TSVDUnfold itself, it calculates a new covariance matrix Bdata, which

is the error matrix for the unfolded distribution. Finally, after training, TSVDUnfold goes

through step 9 in the algorithm and gives the unfolded distributions x and xdata. x is the

unfolded distribution for the reconstructed simulated events, and xdata is that for the real

data events.

Unfolded Distributions


Eve

nts

/ 0.5

GeV

0

100

200

300

400 True Energy Spectra. NDOS MC. No Unfolding.ν

Reco MC

Fake Data

(a)


Eve

nts

/ 0.5

GeV

0

100

200

300

400

True MC

Unfolded Fake Data

True Energy Spectra. NDOS MC. With Unfolding.ν

(b)

Figure C.1 Neutrino Energy Distributions Before And After Unfolding. (a) Recon-structed MC simulation (red) normalized to fake data (black) area. (b) True MC simulation(green) normalized to unfolded data (black) area. MC simulation.

Following the steps presented at the end of the previous section, and in order to train

3The size of this toy MC simulated set is entered by the user. The recommended size isnTSVD = 100 events.

217

the algorithm, the MC simulated sample is divided into two sets: one used to build ~x ini,

training sample, and the other one4 used to build ~b ini, testing sample. The TSVDUnfold

is fed with the due inputs, and the results of the training process are presented in figure

C.1. As presented in the previous section, the parameter k comes from figure C.2b, which

shows that the last statistically significant equation is that with i = 2, since for higher i the

distribution oscillates randomly around |di| = 1. The parameter k is set to:

k = 2, τ = s22. (C.23)

The migration of events from their reconstructed energy bins to their unfolded energy bins

is presented in figure C.2a, which is the covariance matrix Bdata mentioned above. The

diagonal entries between 1 GeV and 2.5 GeV are 20.1% of the total entries. In this case the

correct bin is reconstructed properly. The off diagonal bins in that same energy region, and

colored with light orange, account for 27.2% of all entries.

(GeV)Reco

Energyν0 2 4 6 8 10

(G

eV)

Tru

e E

nerg

yν

0

2

4

6

8

10

0

50

100

Energy. NDOS MC Covariance Matrix. ν

(a)

i0 5 10 15 20

| i|d

-310

-210

-110

1

10

| vs. i. TSVDUnfold.i

log|d

(b)

Figure C.2 Results From Unfolding Fake Data. (a) The covariance matrix Bdata. (b)log|di| vs. i. MC simulation.

Figure C.1 shows 4 distributions: The true MC simulated events, the reconstructed MC

4About 10% the size of the first sample.

218

simulated events, the fake data5 events, and the unfolded events. True MC and Reco MC

simulations resemble figure 6.14. The fake data, as part of the MC simulated sample, should

have the same shape as the Reco MC simulation does, which is the case in figure C.1a.

The unfolded distribution sits on top of the true MC simulation one, which means that the

unfolded procedure worked as expected, as seen in figure C.1b. The errors in the unfolded

distribution come from the input to the covariance matrix, which in this case are statistical

errors in the fake data.

The goal of the unfolding was to place all the fake data events into their most probable

energy bins given the performance of the detector embedded in the calculated covariance

matrices. This corrects for the discrepancies between reconstructed and simulated events

presented in chapter 6.

The training and testing of the unfolding algorithm are satisfactory, and the algorithm

is applied to the NDOS data. In order to find the unfolded data distributions, ~bdata is

filled with real NDOS events instead of fake data events, and the covariance matrix Bdata is

calculated based on the real data measured errors. Everything else is the same as presented

above, since it is the result of training the algorithm.

5The small MC simulated testing sample.

219

Appendix D

List Of Muon Neutrino Charged

Current Candidate Events

Run Event Eν Run Event Eν Run Event Eν

(GeV) (GeV) (GeV)

13067 01 52339 1.1 13067 05 242768 1.3 13084 14 636949 1.413097 09 401527 2.0 13119 19 878982 1.4 13120 06 275972 1.513131 08 369633 2.3 13168 09 439180 2.0 13173 00 21364 1.013178 09 423895 1.0 13209 07 351868 1.5 13210 12 558497 0.813228 00 7822 0.9 13247 01 88704 1.0 13247 08 381991 1.313247 10 468719 1.0 13286 00 31278 0.9 13286 04 185526 1.013289 00 3856 1.8 13352 09 455006 1.5 13352 15 686658 2.113359 03 149922 1.3 13359 18 838462 2.1 13390 19 898936 1.213405 00 21468 2.1 13405 06 283899 2.5 13405 08 373623 2.113405 09 423816 1.0 13406 05 269806 1.1 13406 19 895717 2.013417 14 647470 2.0 13417 40 1815674 1.5 13485 18 898130 0.913504 00 5587 1.6 13508 11 502607 0.9 13508 17 791080 1.413521 07 320076 1.7 13523 09 429603 2.0 13525 00 30494 3.113526 03 173654 1.4 13528 16 704288 0.9 13531 02 100521 1.113531 02 100521 1.1 13540 09 432385 1.0 13543 04 197375 0.913550 01 67262 1.0 13557 20 853445 3.5 13567 00 32484 1.113648 05 224998 1.9 13659 03 154457 2.1 13659 07 333945 1.213659 12 576285 1.2 13675 05 251706 1.7 13679 08 402161 1.913679 12 565281 1.8 13679 13 624373 1.1 13709 18 838184 0.913730 20 948244 1.0 13731 18 858628 1.4 13732 01 83757 1.513732 04 224730 1.4 13732 05 233639 1.0 13732 06 281760 3.113737 02 112479 1.2 13737 04 188755 0.9 13737 08 413309 1.013743 03 176473 1.0 13754 01 62591 1.0 13754 02 123870 2.513756 04 219066 3.3 13759 05 248715 1.2 13763 04 162894 0.613767 01 68409 1.2 13767 02 81374 3.4 - - -

Table D.1 Selected QE Candidate Events.

220

All the selected QE neutrino candidates before cosmic background subtraction are pre-

sented in table D.1. All the selected non-QE neutrino candidates before cosmic background

subtraction are presented in tables D.2 and D.3.


(GeV) (GeV) (GeV)

13068 01 70583 1.9 13068 02 133375 1.1 13068 03 183770 1.813071 05 273089 1.5 13075 15 704141 2.0 13075 17 781439 2.513084 01 87787 1.6 13087 01 57985 3.2 13094 05 265850 2.013097 11 516437 2.4 13097 11 517102 1.8 13097 13 604309 2.013117 03 151666 0.9 13120 00 38220 1.3 13123 04 158468 1.313131 01 73464 2.4 13160 01 61236 1.3 13160 02 112807 2.013160 11 512920 1.5 13160 15 687140 2.3 13166 09 410288 1.313173 12 576650 1.6 13176 04 244242 1.7 13176 11 539526 1.413178 04 206510 1.6 13178 05 266544 2.1 13178 07 323898 4.513189 02 95469 1.5 13190 00 7437 2.4 13208 03 140024 1.913209 01 71489 1.5 13209 14 655627 2.0 13209 19 868527 0.813210 13 608565 3.1 13213 02 94255 1.8 13214 03 177158 1.413220 01 84251 2.2 13236 05 246984 2.4 13241 11 541964 1.113247 01 55167 1.6 13247 09 430374 1.8 13262 01 55638 4.013262 01 70110 1.6 13265 04 229559 2.6 13265 06 292238 1.313277 01 79372 1.9 13286 01 48964 2.3 13286 02 120280 1.513312 06 315859 1.7 13320 15 685032 1.6 13330 02 117981 1.913333 06 285527 1.9 13333 10 488169 1.3 13333 12 546198 2.713333 15 694421 3.0 13339 00 20224 2.2 13339 03 135737 1.913348 06 271122 1.5 13352 05 269563 1.7 13352 10 494823 2.013352 15 706818 1.2 13353 13 611234 1.1 13359 12 558228 1.713360 05 260305 2.0 13365 02 118904 1.7 13365 03 141066 1.313389 02 98800 2.9 13389 11 533620 2.8 13393 07 326670 1.613393 19 880031 2.3 13393 20 933931 1.6 13399 06 300556 2.013399 10 470075 1.7 13399 12 578886 1.5 13399 12 589675 1.413405 06 281676 2.3 13405 18 857049 3.0 13405 20 936462 2.113406 08 396159 2.6 13406 13 641569 1.1 13406 20 945997 1.513407 00 40739 1.9 13407 09 435601 2.1 13409 14 668614 1.813441 06 320584 1.4 13443 09 398666 1.9 13443 14 664933 1.3

Table D.2 Selected Non-QE Candidate Events, I.

221


(GeV) (GeV) (GeV)

13444 09 445873 2.0 13444 13 632045 1.3 13485 14 680500 1.513490 00 27697 2.2 13490 00 43841 1.9 13495 15 714932 2.213504 03 163755 2.9 13504 03 170439 2.8 13505 06 317921 2.413508 00 31211 3.4 13508 12 560534 1.7 13508 13 610384 2.713512 06 292369 2.2 13512 14 667424 2.4 13516 01 55536 1.513521 04 219941 1.3 13521 07 334779 1.7 13523 03 153709 3.713523 12 573584 1.7 13523 22 1005777 1.8 13525 06 275923 1.113531 08 373797 1.3 13533 07 337448 3.3 13533 17 806866 1.613534 00 18678 1.3 13539 03 169453 2.3 13540 20 917878 1.613557 07 261561 2.4 13557 14 619026 2.1 13557 15 669256 1.813611 04 184605 1.2 13619 04 205187 1.1 13619 11 525351 1.513619 18 843304 1.2 13659 11 534603 1.9 13660 05 239565 1.513660 07 338739 1.6 13660 10 493358 2.3 13709 16 746592 1.813710 18 864719 2.2 13711 03 150155 1.5 13711 06 300659 4.113717 12 589572 2.3 13717 15 717932 1.3 13727 13 608348 3.313731 14 688158 2.0 13731 19 917338 1.4 13732 03 151136 1.713732 05 236320 1.3 13737 04 197897 1.4 13737 05 263039 1.513740 00 11613 3.0 13740 21 972374 1.8 13742 05 270585 2.013743 00 7658 1.0 13743 08 406626 1.2 13743 23 1069794 1.813744 04 188689 1.2 13744 05 272594 2.6 13744 07 362545 2.013744 10 495266 1.4 13746 23 1096829 2.6 13751 15 700296 1.513754 01 85788 1.2 13754 12 588024 2.8 13754 15 691009 1.413754 16 773307 1.8 13754 16 821744 1.5 13759 19 888335 0.913760 17 750963 2.2 13763 01 70828 1.9 13763 09 358811 1.013764 00 4051 1.1 13765 03 146213 1.5 13767 02 91379 1.613770 01 43260 2.0 13770 09 344492 2.3 13778 22 831025 1.4

Table D.3 Selected Non-QE Candidate Events, II.

222

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